Method and apparatus for tuning compensation parameters in a motion control system associated with a mechanical member

ABSTRACT

In one embodiment, a method and machine are provided for tuning compensation parameters in a motion control system associated with a mechanical member. The method includes the steps of receiving an indication of a compensation parameter to be tested, based on the compensation parameter to be tested causing a signal associated with a desired motion of the mechanical member to be commanded, acquiring control data associated with the signal, acquiring measurement data associated with actual motion of the mechanical member in response to the signal, analyzing the control and measurement data; and based on the step of analyzing the control and measurement data, implementing a value of the compensation parameter.

RELATED APPLICATIONS

The present application is a Divisional of U.S. application Ser. No.10/124,639 filed Apr. 17, 2002, the entire disclosure of which is herebyincorporated by reference. This application claims the benefit of U.S.Provisional Application No. 60/286,834, filed Apr. 26, 2001, the entiredisclosure of which is hereby incorporated by reference.

TECHNICAL FIELD

The present invention relates generally to motion control systems, andmore particularly, to a method and system for tuning compensationparameters.

BACKGROUND OF THE INVENTION

The performance of a motion control system is of primary importance. Inparticular, a control system should be stable, result in acceptableresponses to input commands, result in a minimum steady-state error forinput commands, and be able to eliminate the effect of undesirabledisturbances. Often-times, for optimum is performance, adjustments aremade to a control system. The alteration or adjustment of a controlsystem in order to provide a desired performance is often referred to ascompensation (i.e., compensation is the adjustment of a system in orderto make up for deficiencies or inadequacies).

In some cases, an additional component (e.g., a compensator) is includedwithin a motion control system to equalize or compensate for suchdeficiencies/inadequacies. For example, such compensators can beimplemented as one or more parameters whose values are used by acontroller to determine how to control a system. Accordingly, to provideoptimum performance, appropriate values are determined for suchparameters.

The process of adjusting these parameters (e.g., to compensate forsystem changes) is often referred to as tuning. Recently, techniques fortuning such parameters that involve minimal or no operator involvementhave been developed. These techniques are generally referred to asautotuning. For example, autotuning can refer to a process in which acontroller automatically determines appropriate values for suchparameters.

Currently, in motion control systems such as those utilized with machinetools, relative displacement sensors, such as a telescoping ballbar, areused to provide an electrical output representing a measurement ofincremental displacement caused by the motion control system. Theelectrical output can then be used to “tune” parameters of the motioncontrol systems that account for errors in the motion of the machine(s)it controls. For example, a ballbar can be mounted between the twomembers of a machine tool that move relative to each other and createfeed motion between a tool and a workpiece. More particularly, withrespect to a machining center, for example, the ballbar is mountedbetween a spindle and a work-holding table.

Typically, the range of motion of a commercially available ballbar islimited to just a few millimeters. For example, the machine tool motionduring a ballbar measurement is conventionally along a substantiallyconstant radius arc. Measurement of deviation of the arc radius from theideal constant radius can be used to provide information about theerrors associated with the machine tool. This information can, forexample, be used to determine certain aspects of positioning (static anddynamic) errors associated with the machine.

With reference to FIG. 1, a conventional use involving a motion controlsystem and a machining center is herein described (where a user of themotion control system manually performs most steps). The center fixtureof a ballbar is placed at a desired location on the machine with thesetscrew loose. The spindle is then moved such that a magnetic cup inthe tool holder of the spindle picks up the steel circle center ball ofthe ballbar and the setscrew is tightened. The positions of the axes arethen recorded by performing a machine position set to redefine thecoordinate system, such that the current position is the program systemorigin (e.g. G92X0Y0Z0).

A user of the motion control system then manually feeds the axes to movethe magnetic cup off of the center ball, and creates a part programcausing the machine to perform a circular move, an exemplary one of thepart programs having a number of characteristics, including thefollowing:

-   -   1. The circle center corresponds to the location of the center        ball.    -   2. The circle radius is equal to or slightly less than the        nominal length of the ballbar.    -   3. The circular motion is constrained to lie within a major        plane of the machine (e.g., XY, XZ, YZ).    -   4. The circular arc includes some additional (overshoot)        distance at the beginning and end of the move (allows the        feedrate to accelerate from zero to full speed and to decelerate        from full speed to zero), with the portion of the circle whose        data is considered by the analysis being referred to as the        data-arc (data collected during the feedrate transients at the        beginning and end of the move is not considered during data        analysis).    -   5. The circular motion is preceded by a short (feed-in) move        that travels along a circle radial line from a radius greater        than the circle radius to the circle start point. Similarly, a        short (feed-out) move following the circular motion travels        along a circle radial line from the circle end point to a radius        greater than the circle radius. These moves can be used to        signal conventional software associated with the ballbar when        the test begins and ends, and can be used by the software to        provide an estimate of the instantaneous circle angle based on        the elapsed time since the test start.    -   6. Two successive sequences with overshoot arcs, a feed-in move,        a data-arc, and a feed-out move are present for both clockwise        and counter-clockwise directions within the plane.    -   7. A program stop (M0) block is present prior to the beginning        of each (clockwise and counter clockwise) sequence of moves (the        program stop providing the operator with an opportunity to arm        the trigger in the ballbar software).

The ballbar can then be plugged into a serial port of the computerrunning the ballbar software, and the ballbar length optionallycalibrated (e.g., by placing the ballbar into a calibration fixture ofknown length and prompting the software to measure the ballbar'soutput). If the ballbar length is calibrated, the actual ballbar lengthis computed and stored by the ballbar software. One advantage of alength-calibrated ballbar is that it enables average circle diametererror to be estimated.

The ballbar software is then informed of the specific conditions for thecurrent test by entering values for the following parameters: length ofovershoot arcs, data arc start angle, data arc length, feedrate, andcircle radius. The trigger in the ballbar software is armed on themachine (e.g., a personal computer) on which it is running, and the testprogram on the control is executed.

When the program reaches the stop (M0) after the first circular move,and after the data collection is completed, the ballbar software triggeris re-armed, and the machine is cycle started to perform the secondcircular move. At test completion, the test data is saved to a file andanalysis software is run to diagnose error sources.

Accordingly, a method and system to automate and simplify this and othersimilar procedures is desired. In addition, it would be desirable toapply parametric compensation based on measurements from a ballbar.

SUMMARY OF THE INVENTION

It is an object of at least one embodiment of the present invention toprovide an improved method and/or system for tuning compensationparameters of a motion control system.

Additional objects, advantages, and novel features of variousembodiments of the invention will be set forth in part in thedescription of the exemplary embodiments that follows and, in part, willbecome apparent to those skilled in the art upon examining or practicingthe invention.

One embodiment of the present invention is directed towards a a methodfor tuning compensation parameters in a motion control system associatedwith a mechanical member. The method includes the steps of receiving anindication of a compensation parameter to be tested, based on thecompensation parameter to be tested causing a signal associated with adesired motion of the mechanical member to be commanded, acquiringcontrol data associated with the signal, acquiring measurement dataassociated with actual motion of the mechanical member in response tothe signal, analyzing the control and measurement data; and based on thestep of analyzing the control and measurement data, implementing a valueof the compensation parameter.

Still other advantages of various embodiments will become apparent tothose skilled in this art from the following description wherein thereis shown and described exemplary embodiments of this invention simplyfor the purposes of illustration. As will be realized, the invention iscapable of other different aspects and embodiments without departingfrom the scope of the invention. Accordingly, the advantages, drawings,and descriptions are illustrative in nature and not restrictive innature.

BRIEF DESCRIPTION OF THE DRAWINGS

While the specification concludes with claims particularly pointing outand distinctly claiming the invention, it is believed that the same willbe better understood from the following description taken in conjunctionwith the accompanying drawings in which like reference numerals indicatecorresponding structure throughout the figures.

FIG. 1 is a diagram illustrating a ballbar test;

FIG. 2 is a screen shot of a setup interface according to an exemplaryembodiment of the present invention;

FIGS. 3A and 3B are screen shots of a test interface according to anexemplary embodiment of the present invention;

FIGS. 4 and 4A are screen shots of an analysis interface according to anexemplary embodiment of the present invention;

FIGS. 5A-5E are screen shots of various subpages of an adjustmentinterface according to an exemplary embodiment of the present invention;

FIGS. 6A and 6B are plots illustrating formulation of circle centererror and the polar error plot signature resulting from a circle centererror;

FIG. 7 is a plot illustrating an effect of squareness error on radius;

FIG. 8 is a plot illustrating effects of axis scale errors;

FIG. 9 is a block diagram illustrating assumed axis position loopdynamics for estimating radius distortion as influenced by position loopband width according to an exemplary embodiment of the presentinvention;

FIG. 10 is a polar error plot illustrating an effect of a parabolicstraightness error;

FIG. 11 is a plot illustrating an effect of servo (phase) mismatcherrors;

FIG. 12 is a plot illustrating an effect of lost motion on circularcontour;

FIG. 13 is a polar error plot illustrating a lost motion signature;

FIG. 14 is a polar error plot illustrating a friction reversal errorsignature;

FIG. 15 is a block diagram illustrating a system model employed byfriction error estimation according to an exemplary embodiment of thepresent invention;

FIG. 16 is a graph illustrating excitation as a function of data arclength according to an exemplary embodiment of the present invention;

FIG. 17 is a plot illustrating an example of angular error (y-axis rollerror);

FIG. 18 is a graph illustrating clipping behavior of appliedstraightness compensation;

FIG. 19 is a graph illustrating maximum and minimum axis deflectionbased on start and end quadrant according to an exemplary embodiment ofthe present invention;

FIG. 20 is a block diagram illustrating a multi-threaded implementationof test realiziation according to an exemplary embodiment of the presentinvention;

FIG. 21 is a block diagram illustrating resampling according to anexemplary embodiment of the present invention;

FIG. 22 is a legend explaining symbols used in finite state machinediagrams according to exemplary embodiments of the present invention;

FIG. 23 is a finite state machine diagram for a run test coordinator(RTC) algorithm according to an exemplary embodiment of the presentinvention;

FIG. 24 is a diagram explaining a reusable “A” state according to anexemplary embodiment of the present invention;

FIG. 25 is a diagram explaining a reusable “W” state according to anexemplary embodiment of the present invention;

FIG. 26 is a plot illustrating machine motion for backlash testcounterclockwise arc according to an exemplary embodiment of the presentinvention;

FIG. 27 is a class diagram illustrating a data hierarchy for a backlashtest data structure according to an exemplary embodiment of the presentinvention;

FIG. 28 is a flowchart illustrating a backlash program generationprocedure according to an exemplary embodiment of the present invention;

FIG. 29 is a finite state machine diagram for a test in progress (TIP)algorithm according to an exemplary embodiment of the present invention;

FIG. 30 is a finite state machine diagram for a ballbar lengthcalibration algorithm according to an exemplary embodiment of thepresent invention;

FIG. 31 is a diagram illustrating structural dynamics parameterestimation using a ballbar according to an exemplary embodiment of thepresent invention; and

FIG. 32 is an abstract framework for auto tuning on a CNC according toan exemplary embodiment of the present invention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

In general, the present invention relates to a motion control systemcapable of tuning compensation parameters. For purposes of illustration,exemplary embodiments discussed primarily herein relate to test softwarethat runs on the control of a machine tool and, in one embodiment, tosuch software that allows a machine tool control to self-calibrate itsparametric compensation based on measurements from an instrument calleda telescoping ballbar. Moreover, in one embodiment, disclosed exemplarysoftware provides the capability to: 1) specify test conditions; 2)automatically create the motion for measurements related to the testcondition; 3) initiate the motion; 4) collect the data; 5) analyze thedata; and 6) update the compensation of the control based on the data.

Further exemplary embodiments can, for example, be used to:

-   -   1. Validate the appropriateness of test setup data.    -   2. Validate the integrity of collected data.    -   3. Aid a user in fixture setup by providing prompts of what to        do next based, for example, on control state and axis positions.    -   4. Monitor a control state to coordinate a test sequence.    -   5. Send messages to control the execution of part programs.    -   6. Robustly handle error conditions.    -   7. Process measured data to account for effects such as servo        gains and feedforward.    -   8. When computing proposed changes to compensation, consider the        values for compensation that were active during the test        execution.    -   9. Copy updated compensation values into, for example, a        database of a control.    -   10. Activate new compensation (e.g., by issuing a system event).        For example, certain exemplary embodiments of the present        invention make use of access to controller internal data,        program buffer, and system event services of an associated        control to provide at least some of the aforementioned features.

According to one embodiment of the present invention, configuration ofcertain types of compensation can be automated. Referring specificallyby example to an exemplary embodiment involving ballbars, values for thefollowing types of compensation for axes in a machine tool that define aprimary machine plane in which a ballbar circle test is performed can beautomatically proposed: scale error; squareness error; straightnesserror; backlash; windup, and/or friction.

Ballbar measurements of radius deviations that occur during major planecircular motion of a machine are the combined result of the simultaneousmotion of two axes. In order to attribute the measurements to errorsources from a single axis, the analysis can be restricted based on aseries of limiting assumptions about the nature of the errors. Forexample, the errors can be assumed to originate from a limited number ofsources with a fixed mathematical relationship between an error source,the axes positions, axes velocities, and the radial error produced.

The particular mathematical relationships included within the model canbe chosen based on some physical understanding of the machine structureand servo dynamics, and/or on previous experience with machine toolerrors. Accordingly, each error source can be described by a singleparameter that quantifies the amplitude and direction of the error.Therefore, data representing such error can be curve fit to extractoptimal estimates for each parameter associated with each error type.

An alternative to the parametric model could include a data-basedapproach. If, for example, the desired goal of the ballbar measurementwas the generation of data tables for bidirectional compensation(compensates for backlash and error due to variation in the pitch of aballscrew) and cross-axis compensation (compensating for axis positionas a function of one or more axis positions), then there might be aninfinite number of data set combinations for these tables that mightexactly account for the errors measured by the ballbar. However, onlyone of these combinations would be the “correct” one. Meanwhile, anincorrect selection could easily result in larger errors than what mightoccur on an uncompensated machine (e.g., for any axis motion thatdiffers from the motion experienced during the ballbar measurement).

Although a parametric description of the errors might not exactlydescribe the measurement, it can allow for the derivation of an optimalcombination of parameters that best describes the measurement.Accordingly, association of measured machine errors to a parametricmodel implies that the compensation for the machine errors can be basedon the same parametric model.

In, for example, an A2100 control from Siemens Energy and Automation,Inc., the following types of parametric compensations are available: 1)backlash compensation; 2) windup compensation; 3) pitch (or scale)compensation; 4) squareness compensation; and 5) straightnesscompensation. Each of these compensations describes the motion of anindividual axis or combined effect of two axes within a plane. Each ofthese error sources can also be described by a single parameter.

A mathematical model can be based on an understanding of the physicaleffects that often cause machine errors. If an error source exists thatis outside of the scope of the mathematical model, then the error, mightbe falsely attributed to one of the error types that is included in themodel structure. To help avoid this situation, the scope of themathematical model can be made larger than the scope of parametriccompensations provided.

For example, servo mismatch error can be included within the scope ofone of the exemplary models described herein for use with the A2100control, even though the A2100 control does not have an explicit servomismatch compensation. Such servo mismatch errors can instead becorrected in other ways based on other types of measurements. In anycase, situations might arise where the mathematical model cannotproperly describe the data. For example, this could occur if a machinehas an error that is outside of the scope of the particular mathematicalmodel, if there are problems with the measurement, such as the ballbarseparating from the magnetic cup, or if there are high levels of noisein the measurement. To identify such a situation, statistical measuresof the coherence between the model and the measurement can be computedand, for example, displayed to a user of the system (e.g., where a lowvalue for coherence indicates to the user that a closer look at the datamight be required).

According to one embodiment of the present invention, parameterestimates are used to describe machine errors within the range of motionthat took place during a ballbar test. One potential concern associatedwith extrapolation of the model to regions outside of the range ofmotion of the test involves the uncertainty of the parameters in theseregions. Therefore, one embodiment of the present invention includes alimitation that it reliably compensates for the region of a plane thatis within the range of motion of the ballbar test. For example, formachine planes that have a rectangular (rather than square) range ofmotion envelope, a single circle test might not produce compensationsthat are reliable over the full range of motion.

Furthermore, certain error types appear to be more reliably extrapolatedthan others. For example, backlash, windup, and scale errors appear tobe much more uniform over the full range of the axis than a parametricsquareness and straightness error. Moreover, parametric compensation ofa machine may be inappropriate in situations where the range of motionof the ballbar test used to develop the compensation is significantlyless than the range of motion of one or more of the machine axes. Insuch a situation, parametric compensation based on a ballbar test may bemost useful as a supplement to data-based bidirectional-axis-(pitch)-compensation and cross-axis-compensation that covers the fullrange of axis motion.

In other embodiments involving a ballbar test, the test may be used bythe end user to fine-tune the compensation over a limited range of axismotion that represents the range of motion to be employed in animmediately upcoming machining operation. Accordingly, in one embodimentof the present invention, it is recommended that the diameter of thecircle test encompass as much of the machining envelope as possible. Ascan be understood, the ability of the parametric model to exactlydescribe the motion within the limited region increases as the range ofmotion (circle diameter) decreases, (the error types are more likelywithin the scope of the model).

Another consideration that might affect the reliability of theparameters can be the nature of the circular motion itself. For example,sufficiently exciting each error source such that it produces a uniqueeffect on the measurement allows different error sources containedwithin a model to be distinguished. For an exemplary model describedherein, sufficient excitation involves collecting data for bothclockwise and counter-clockwise moves. Additionally, in a furtherexemplary embodiment, the length of the data arc is as large aspossible.

In an exemplary embodiment, both clockwise and counter-clockwise dataarcs cover 360 degrees. In some cases, however, the mechanicallimitations of the machine might not allow for a test thatsimultaneously covers a 360-degree arc and the maximum machiningenvelope of one or both of the axes. In such a situation, one embodimentof the present invention might involve using a data arc that is lessthan 360 degrees to enable the full axis envelope to be covered. In anexemplary embodiment described herein, the data arc is, however, atleast 180 degrees, and the clockwise arc covers the same angular rangeas the counter-clockwise arc.

A statistical value referred to herein as the excitation may also becomputed to provide the user with an indication of the sufficiency ofexcitation for distinguishing effects of different parameters. Two ofthe exemplary parametric errors, backlash and windup, are typicallyindistinguishable from each other during a standard (continuous motion)ballbar test, which provides an indication of the total lost motion. Inone embodiment of the present invention, a separate backlash test isperformed to determine the contribution of the backlash error to thetotal lost motion indicated by the measurement data from continuousmotion test.

According to an exemplary embodiment of the present invention, a userinterface, such as one containing four primary modes, can be providedfor the system. In one embodiment, these modes include a setup testmode, a run test mode, an analyze data mode, and an adjust compensationmode. For example, each mode can be displayed as a separate page on auser interface.

Referring now to FIG. 2, in one embodiment of the present invention, atest-setup page 10 can be provided that acts as an interface for a userto, for example, either load a set of test condition parameters from afile or to specify such parameters manually. In an exemplary embodiment,the test conditions specified on this page can be used (e.g., by thecontrol) to automatically generate a part program during a testexecution stage (such as under the run-test mode that will be laterdescribed herein).

According to one exemplary embodiment, the test condition parameters maybe specified for one test in each major plane of an associated machine(e.g., XY, XZ, YZ). For example, the user may modify the test conditionparameters by entering an edit mode. While in edit mode, fields 12 thatcontain the test condition parameters could become modifiable.

Once any desired changes have been made, the user can then press an“apply” button 14. In such an embodiment, pressing an apply button 14can be used to initiate a series of validation checks for the enteredtest condition parameters. For example, validation checks could be usedto ensure that a reasonable test may be developed from an enteredcombination of test condition parameters. In one exemplary embodiment,if a validation test fails, an alert describing the problem isdisplayed, the user interface remains in edit mode, and the user isexpected to change the test parameters until they represent a validcombination.

In a further exemplary embodiment, a separate group of test conditionparameters can be stored for each of the three major planes. The testcondition parameters could include the following:

-   -   1) circle radius (less than or equal to the ballbar length)    -   2) circle center (location in X, Y, Z where circle should be        centered)    -   3) data arc start angle (counter clockwise angle from horizontal        axis at which to begin data collection)    -   4) data arc length or test range (arc angle over which data is        collected);    -   5) overshoot arc length (arc angle where data is not collected        to allow for transients during acceleration and deceleration; in        one embodiment the acceleration overshoot arc and deceleration        overshoot arc can both be specified by this parameter)    -   6) ballbar length (this selection might be limited based on        available extension bars);    -   7) feedrate (programmed feedrate for data-arc, overshoot-arcs,        feed-in, and feed-out); and    -   8) process or feed mode (specifies the group of path parameters        (e.g., G45.n) to be used for motion during the test; this may        affect path acceleration, path jerk, step velocities, and        feedforward status)        In one embodiment of the present invention, a combination of        test conditions parameters are validated, such as against the        following: 1) Is circle radius appropriate for the ballbar        radius; 2) Is axis motion within the machine range?; 3) Is        circle center within the machine range?; 4) Are overshoot arcs        sufficiently long to enable constant feedrate during data arc?;        and 5) Are the data arcs long enough to provide sufficient        excitation to distinguish between error types?

In an exemplary embodiment of the present invention, a circle radius isrequired to be less than or equal to a ballbar radius. According to suchan embodiment, if the circle radius is less than the ballbar radius, thelength of the ballbar traces a conical surface. In such an embodiment,the circle radius is slightly less than the ballbar radius because themeasurements might become overly sensitive to motion in the direction ofa third axis (e.g., the axis perpendicular to the major plane). Motionalong the direction of the third axis should be substantiallyindistinguishable from motion of the axes in the plane and can oftendistort the parameter estimation.

For example, as the angle of the cone increases from 0° (circle radiusequal to the ballbar radius) to 45°, the relative contribution toballbar deflection from motion in the third axis increases from 0% to100% of the deflection caused by motion in the plane. A respectiveexemplary validation check requires that the cone angle not exceed 26°.In such an embodiment, at 26°, the relative contribution to ballbardeflection from motion of the third axis is 50% of that of motion withinthe plane.

The selected values for ballbar length, circle radius, circle center,overshoot arc, data arc start angle, and data arc length can also beevaluated to ensure that the motion produced is everywhere within therange limits of the axes. For example, if it is detected that an axisrange limit might be exceeded, a validation check can provide proposalsfor changing, for example, the circle center to allow the circle to fitwithin the envelope of the machine (if the motion can be brought towithin the machine envelope by shifting the center).

Even if the circular arc is within the machine's range, it can still bepossible that the circle center is not within the range of the machine.For example, this can occur when the circle radius is less than theballbar length. Accordingly, in one embodiment, as a tool tip (e.g., amagnetic cup) should be brought to a circle center ball to locate acentering fixture, the circle center should be within the range of themachine.

Another exemplary embodiment can include requiring collection of dataduring a period of time when the circle is traversed at a constantfeedrate. The frequency response of an axis typically involvesdistortion of the amplitude and phase as a function of frequency (for acircle, the frequency is proportional to feedrate). At a singlefeedrate, the amplitude and phase distortion should be constant, theshape of the circle should not be distorted (from individual axisfrequency response), and it should be possible to compensate for theeffects due to constant amplitude attenuation and phase lag when postprocessing the data.

An acceleration overshoot arc can be included to help ensure that themachine axes have an opportunity to accelerate to full feedrate beforedata collection begins. Meanwhile, a deceleration overshoot arc can beincluded to avoid collecting data during the deceleration from fullfeedrate to a full stop. While, in one embodiment, the overshoot arcsshould be long enough to ensure constant feedrate during the data-arc,it can also be desirable to keep the length of the overshoot arc to beonly slightly above its minimum possible value to minimize the totaltest time.

Length of the acceleration overshoot arc can depend on several factors,including: 1) an end of span speed constraint of a feed-in spanpreceding the acceleration overshoot arc, 2) a span maximum accelerationand jerk for an overshoot arc span, 3) programmed feedrate, 4) axis Kvgains, 5) axis feedforward levels, and/or 6) whether rate feedforward isactivated. Similarly, the length of the deceleration overshoot arc candepend on similar factors. Furthermore, the span data's acceleration,jerk, end-of-span-speed-constraint, and feedforward status can varydepending, for example, on the axis configuration data, pathconfiguration data, feed mode selection, data-arc start angle, and dataarc length.

Typically, lengths for the acceleration and deceleration overshoot arcscan be substantially identical. In one embodiment, whether an overshootarc is acceptably long can be determined by running a sequence of spansin a test move through motion algorithms of the control without directlycontrolling the machine. In an exemplary embodiment, appropriatesubsystems of motion algorithms of the control (e.g., look-ahead andvelocity management) are linked to form a limited scope emulation thatcan run as a callable function in a non real-time environment. Themotion can be emulated in much less time than the actual move by, forexample, limiting the emulation to small portions of the movement withthe correct initial conditions (velocity and acceleration).

The emulated path distance signal can be passed through a filteringfunction that provides the same phase and amplitude distortion createdby the axis Kv and feedforward level. According to one embodiment of theinvention, the results of the emulation can be used to indicate whetheror not the overshoot arcs are long enough. If the arcs are not longenough, then a conservative approximation can be used to develop aproposed value for overshoot arc length to provide to the user. This newestimate can also be validated against the actual motion.

With respect to one exemplary embodiment, an estimate of total lostmotion for both axes in a plane can be obtained by including at leastone reversal for each axis during the motion. This implies a minimumrange of motion of somewhat greater than 90° if the data is aligned withone of the circle quadrants. In order to accommodate any data arc startangle, data arc length can be set somewhat greater than 180° to helpensure the occurrence of at least one reversal for each axis.Additionally, 180° of motion can help afford the ability to reasonablydistinguish between squareness and straightness errors. Thus, in such anexemplary embodiment, a minimum allowable data arc is 180° and anexcitation parameter exceeds a minimum defined threshold.

Setup test page 10 can also add additional content to increase overallusability. For example, a setup test page 10 can include a “Get Current”button 18 that, when pressed, copies current machine coordinates of theaxes to the circle center coordinates for the current plane. It can alsocontain an “Auto Select” button 20 that sets the circle center as themiddle of the machine's maximum range envelope and sets the circleradius equal to the ballbar length. In one embodiment, the “Get Current”and “Auto Select” buttons 18, 20 are enabled when in edit mode.

Setup test page 10 can further contain a graphical and/or animatedrepresentation 22 of the data arc and/or overshoot arc. Additionalcapabilities could include an ability to revert to the original testcondition parameters that were present prior to any changes that weremade during the current edit session (e.g., such as from within an editmode). It can also include the ability to save the current testcondition parameters (e.g., for each of three planes) to a file and/orto load new test condition parameters (e.g., for three planes) from afile. Still further, setup test page 10 can include the ability toassign the currently active test condition parameters as the defaultsetup that is present when the user interface is opened.

In certain embodiments, setup test page 10 can contain content to assigndata associated with, for example, a ballbar device itself. For example,a serial number of the ballbar can be copied to the header of ameasurement data file for the purpose of creating a permanent record ofthe particular ballbar used to generate that data set. A ballbar scalefactor can be used to adjust the measurements to account for the uniquecharacteristics of the specific ballbar (typically printed on the sideof the ballbar). Still further, a COM port to which the ballbar isattached can be specified on this page. There can also be a button thatis used to debug ballbar problems.

Referring now to FIG. 3A, a run-test page 30 can be included, such asone that provides an interface to aid a user in setting up a fixture(e.g., a ballbar) and executing a test (e.g., a ballbar test). Forexample, such a run-test page 30 can comprise a drop-down list 32 forselecting a test (e.g., of three major plane tests) to perform. In oneembodiment involving a ballbar test, for each of the major planes, fivesequential tasks (or visible states) in the overall process of obtainingand saving a measurement are provided.

According to such an embodiment, when a major plane is selected, testsetup data associated with that major plane is employed in the executionof the test. If a new major plane is selected, the state returns to itsstarting point in the test sequence. A finite state machine (FSM) thatrecurrently polls control and acquired data can coordinate theprogression through these states. For example, such a test coordinationFSM can contain more states than those that are made visible as a task.A detailed discussion of one embodiment of such a FSM involving aballbar test is further discussed later herein.

According to an exemplary embodiment of the present invention, progresstowards the completion of the tasks is indicated with a display on ascreen. For example, each task can have an associated checkbox 34 thatbecomes checked when a corresponding task has been completed. A user canalso be provided with the ability to, at any point in the process, goback any number of tasks by, for example, pressing a button associatedwith that task. Additionally, if a task fails to complete properly, theability to automatically back up to the nearest previous state or taskpossible can also be provided.

In one embodiment, throughout a sequential process, a user can beprovided with a text description of what to do next to progress throughthe sequence. For example, control data can be used to determine whenthe requirements for a state transition to the next state in thesequence (of tasks) have been fulfilled. In an exemplary embodiment, auser can avoid the manual intervention involved in completing theprocess by performing all operations using, for example, a pendant orpanel of the control. One advantage associated with such an exemplaryembodiment could include allowing the user to accomplish tasks at amachine without having to return to the control.

In one embodiment of the present invention involving a ballbar test,primary tasks in the process of completing the ballbar test include 1)MoveToCenter, 2) ConfirmCenter, 3) MoveToStart, 4) perform the test, and5) save data associated with the test. A high level description of eachof these tasks according to an exemplary embodiment is given below.

The state associated with the MoveToCenter task can be entered by, forexample, pressing a Move to Center button 36 on run-test page 30.According to one embodiment, when the MoveToCenter state is entered, therecurrent polling for transition stimuli begins (if it has not alreadybegun). Accordingly, when the MoveToCenter state is entered, a sequenceof part program blocks to cause the axes to move to the circle center iscreated and loaded into, for example, a part program Manual DataInterface (MDI) buffer. A prompt text 38 can then be updated to informthe user to, for example, ensure that the path to the circle center isfree of obstructions and then hit cycle start to begin the move. When itis detected that the final block in the move-to-center program has beencompleted and the control-state has transitioned from in-cycle to at endof program (eop) (described later herein), the FSM can transition tostates associated with a ConfirmCenter task.

At the ConfirmCenter state, the FSM cad update prompt 38 to ask the userto, for example, locate the center ball in the magnetic cup of the toolholder, tighten the set screw, and hit cycle start. Upon enteringConfirmCenter, the MDI program buffer can be loaded with some non-motionblocks and dwell blocks whose purpose is to message the FSM as theblocks are executed. Accordingly, when the FSM detects that the user hashit cycle start to confirm the center, the FSM can capture the currentaxis positions and store them as the circle center for the current test.

One advantage of such an embodiment can include facilitating the use ofa slightly different circle center than the one defined in the testsetup data. This can be useful because it can allow the test setup datato contain values for the circle center coordinates that are slightlyoffset from the plane of the center ball. Accordingly, the move tocenter can be performed with a high degree of confidence that the tooltip will not collide with the center ball. When the center has beenconfirmed, prompt 38 can be updated to, for example, suggest that themagnetic cup be power-fed or handwheeled off of the center ball.

The state associated with the MoveToStart task can be entered (e.g.,from the ConfirmCenter task) when the FSM detects that the axes havebeen moved off of the center ball. As the state associated withMoveToStart is entered, the MDI buffer can be loaded with program blocksto move the axes to the start position for the test move and prompt 38can be updated to, for example, suggest ensuring a safe path to thecircle start point and hitting cycle start. The program blocks can movethe axes to the position from which the initial feed-in move originates.

In one embodiment, when the FSM detects that the move to the startposition has completed, it can generate a continuous motion ballbar testprogram and load it into, for example, the MDI buffer. If the user hadindicated that a backlash test was also to be performed, the FSM canalso include backlash test blocks in the program loaded into the MDIbuffer. The display prompt can be updated to suggest that the user placethe ballbar into the machine and hit cycle start to begin the test.

When the FSM detects that the ballbar test program execution has reacheda certain point (e.g., prior to any motion), the FSM can cause programexecution to temporarily suspend (e.g., using a technique to bedescribed in a later section) and attempt to initialize the ballbardevice. If the ballbar is successfully initialized, a button (e.g.,“Cancel Test” button 40 of FIG. 3B) to enable the user to manually abortthe test can be created and displayed, program execution can be allowedto continue, and the FSM can begin polling the ballbar position todetect the initial feed in the first (e.g., counter-clockwise) move.

After detecting initial feed-in, the FSM can begin, for example, storingballbar measurements, checking for error conditions, measuring elapsedtest time (e.g., based on the number of periodic ballbar measurementsobtained), and testing for a feed-out move. In an exemplary embodiment,the program becomes suspended after the feed-out move. When the feed outmove occurs, the data arc can be extracted from the total data set andvalidity checks (to be described later) can be performed. Assuming thatthe validity checks indicate no problems, the FSM can release theprogram from its suspended state and prepare to detect the initialfeed-in from the second (e.g., clockwise) move.

In an exemplary embodiment, the sequence of events that occur during thesecond move is analogous to that of the first move. In one embodiment,once the second move completes and the validity check indicates asuccessful test, the program can begin to execute a backlash test(described later). Once testing is complete and validity checks aresuccessful, the data can be extracted and stored in, for example, atemporary file.

A save data task can be initiated by, for example, the user pressing asave data button 42 after a valid test has been completed. In anexemplary embodiment, save data button 42 is enabled when a test hasbeen completed and validated. For example, pressing save data button 42can bring up an interactive dialog box (e.g., a window) used to assign aname for the file and to provide the user the opportunity to enter otherinformation pertinent to the test (such as the machine type, serialnumber, etc). The data file can contain raw data and a header todescribe the test conditions. In one embodiment, the save dialoginitially contains a default file name that is created based on, forexample, the test plane, machine serial number, calendar date, andnumber of data files with the same properties already present on thedisk.

From the perspective of a user, the performance of a ballbar testaccording to an exemplary embodiment of the present invention could havethe following sequence. The user selects a test plane from a drop downlist 32, indicates a move to center (e.g., via button 36) on a userinterface and indicates cycle start (e.g., via a pendant). An associatedmachine moves to a circle center.

The user manipulates a center fixture and uses a handwheel to make smalladjustments to axis positions until a magnetic cup “picks up” a centerball. At that point, the user tightens a setscrew of the center fixtureand hits cycle start. The user power feeds the axes off of the centerball, hits cycle start and the axes move to a start position. The userplaces the ballbar in the machine, hits cycle start, and the machineperforms the test. When the test is complete, the user indicates thatthe data should be saved (e.g., via a save-data button 42) and the datais stored into a file.

In addition to its ability to perform the aforementioned steps, run-testpage 30 can also provide some additional capabilities. For example,run-test page 30 can include an interface 44 for performing calibrationof a test device (e.g., a length of a ballbar). Run-test page 30 canalso provide an ability to select whether, for example, a backlash testis performed in addition to a continuous motion test. In still furtherembodiments, run-test page 30 can provide the ability to performmultiple tests at, for example, multiple feedrates without having torepeat the move-to-center, confirm-center and move-to-start tasks, andwithout having to return to, for example, setup-test page 10.

As previously alluded to, a length calibrated ballbar can enable adetermination of an absolute axis scale error (a scale mismatch betweenaxes can also be determined, even if the ballbar length is notcalibrated). Accordingly, in one embodiment of the present invention,run test page 30 contains an interface 44 that enables devicecalibration, such as ballbar calibration. For example, such an interface44 can include a numeric field 46 where the user can enter the length ofa calibration fixture, an indicator (e.g., display 48) of whether theballbar is currently length-calibrated, and a button 50 to request thatcalibration begin.

In an exemplary embodiment involving a ballbar calibration, to calibratea ballbar length, a user first enters a calibrator length (e.g., asprinted on the calibration fixture). The entered value is checked (e.g.,by software) to ensure that it is reasonable given the configuredballbar nominal length. Whenever a valid new calibration length isentered, an internal state can transition from calibrated touncalibrated. At that point, display 48 can indicate that the ballbar isnot calibrated and the user can press calibration button 50.

According to one embodiment, pressing button 50 initiates a sequencewhere the ballbar initializes, powers on, collects data for a shortperiod, and powers off. The data can then be checked for constancy andthe reading used to determine an offset to add to the ballbarmeasurement to obtain an absolute ballbar length reading. This value canbe stored and used when processing later measurements. Lengthcalibration can be performed, for example, during the primary tasksdescribed above, with the exception of the perform-test task. In anexemplary embodiment, the test coordinator FSM disables the calibrationcontrols when the state is inappropriate for calibration.

With respect to ballbar testing, a backlash test can be an additionaltest. For example, a backlash test might be used to distinguish betweenwindup and backlash components of the total lost motion measured by acontinuous motion test. The term backlash is used herein to describe atype of lost motion that continues to persist after an axis has come toa stop. The term windup is used to describe a type of lost motion thatoccurs while an axis is in motion.

In an exemplary embodiment implementing a backlash test, the user canindicate that the performance of a backlash test is desired by checkinga “backlash test” check box 52 at some point prior to the move to starttask. Such a backlash test can be used, for example, to measure thepersistent lost motion of each axis in a plane by performing ballbarmeasurements before and after single axis moves that end in a full stop.One exemplary backlash test part program is operative to move the axesalong arcs to each of four circle quadrant boundaries (that are withinthe range specified in setup data). At each quadrant boundary, the partprogram and FSM can effect a sequence of events by incorporating schemesfor messaging between the program and FSM.

In an exemplary embodiment, at each quadrant boundary, the followingsequence takes place:

-   -   1. Perform a single axis move toward the circle center along a        radial.    -   2. Stop axis motion, pause, and collect a first ballbar        measurement.    -   3. Feed the axis further toward the circle center along a        radial.    -   4. Stop axis motion, pause, and collect a second ballbar        measurement.    -   5. Feed the axis back towards the previous point along the        radial.    -   6. Stop axis motion, pause, and collect a third measurement.

According to one embodiment of the present invention, for every ballbarmeasurement, a validity check is performed to ensure that the ballbar isat a constant deflection for the duration of the measurement.Accordingly, the difference between the first and third measurement isthe axis backlash. The second measurement can be used to check validity.The average measured backlash for each measurement point for an axis canbe stored as the axis backlash. Meanwhile, axis windup can be computedby, for example, subtracting the axis backlash from the estimate of theaxis's total lost motion from the continuous test's data.

Run test page 30 can also contain a numeric field 54 that enables theuser to modify a feedrate. Such an embodiment might be useful in that itcould allow the user to run multiple tests without having to return to,for example, test setup page 10. For example, each time the feedrate ischanged, a validity test on the length of the overshoot angles can beperformed. If the overshoot angles are not sufficiently long, an alertmessage can be displayed with a suggested value. In an exemplaryembodiment, to perform a new test after having just successfullycompleted a test, the user can simply enter a new feedrate (if desired)and press a perform-test button 56. In a further exemplary embodiment,each successive test is stored in its own data file, such that everydata file contains, for example, exactly one counter-clockwise resultand one clockwise result.

With respect to the performance of a validity test on data acquiredusing a ballbar, such a validity test can be used to compare a number ofdata points collected through, for example, a data pipe of the ballbar,between a feed-in move and feed-out move to an expected value. If thenumber of data points deviates from the expected number of data pointsby more than a tolerated amount, the total test time is different thanexpected and the data can be considered invalid. An advantage associatedwith such a validity check can be that, if there is a significantdeviation between the expected time and actual time, the estimate ofcircle angle based on elapsed time might not be reliable and thereliability of the data analysis might be-questionable.

Among the many conceivable situations that can cause the actual testtime to differ from the expected time, are: 1) incorrect feedrate, 2)circularity or center-offset errors that are so large that thedeviations cause a stop-trigger to occur in the middle of the circle,and 3) lightly damped servo dynamics that result in overshoot orundershoot during the feed-in and feed-out moves. The expected totaltime between feed-in and feed-out triggers can be determined in advanceby, for example, using motion control subsystem emulation (e.g., similarto the look-ahead/velocity manager/Kv emulation previously mentioned).The tolerance for comparing the measured time to the estimated time canbe based on the uncertainty about the initial ballbar deflection priorto the feed-in move and the final ballbar deflection following thefeed-out move. In one embodiment, the time to execute the emulation maybe minimized, as it measures the elapsed time and distance coveredduring acceleration transients. The expected total time can also be usedby, for example, the test coordinator FSM during test execution todetect an error condition where a feed-out trigger failed to occur whenexpected.

Run-test page 30 can also include numeric displays that indicate axispositions and, for example, an instantaneous ballbar measurement. It canalso contain a graphical display 58 capable of indicating, for example,an instantaneous ballbar location. Such a display 58 might be useful fortracking the progress during the motion. Still further, anotherembodiment of run-test page 30 can include a display 60 of the partprogram blocks (e.g., those currently in a MDI buffer).

Referring now to FIG. 4, an analyze-data page 70 can be used to providean interface for a user to view and manipulate an analyzed measurement.For example, a raw measurement can be stored in a data file thatcontains a header that describes the test conditions and relevantmachine configuration at the time of the test. Analyze-data page 70 canhave a file load/browse interface 72 that enables the user to load andperform analysis on a previously developed data file.

In an exemplary embodiment, when a test run is successfully completed, adata file developed during the run is automatically loaded, so that whenthe user switches from, for example, run-test page 30 to analyze-datapage 70, the recently collected data is already loaded and analyzed. Inone embodiment, whenever a raw data file is loaded, the data undergoesanalysis processing. Data manipulation tasks performed for a file loadaccording to one embodiment of the present invention involving ballbartesting are shown in the following list, and described further herein:

-   -   11. Convert ballbar deflection to circle radius deflection (for        cases where the circle radius differs from the nominal ballbar        length).    -   2. Estimate the circle angle for each data point.    -   3. Estimate the circle center offset and circularity error        (center offset errors could have been created when the center        set screw was tightened, or by a servo drift).    -   4. Estimate the error parameters and their statistical        significance.    -   5. Compute the model coherence statistic.    -   6. Compute the relative contribution of each error type to the        overall circularity error.

In an exemplary embodiment, some of the contents of analyze data page 70associated with ballbar testing include: 1) displays 74 to reflect thetest conditions reported in the file header, 2) an indication 76 of thecenter offset error, 3) an indication 78 of the circularity error, 4) apolar-error plot 80 of a center-corrected measurement, and 5) a legend82 to identify traces on the polar-error plot 80. Polar error plot 80can display, for example, the radius deviation from a reference circleas a function of circle angle. In one embodiment, error amplification isspecified as the plot scaling (e.g., the number of microns per radialdivision on the polar plot's axes). The trace on polar plot 80 can beoffset from the polar origin by some amount to make the plot readable.In an exemplary embodiment, the offset is specified as the percent ofplot full range that defines the radius on the axes that shouldcorrespond to the average radius of the center-corrected measurement.

In still further exemplary embodiments, analyze-data page 70 containsslider controls 84, 86 and numeric-field edit boxes 88, 90 to enable auser to manipulate the scaling and offset used to generate the plot. Thepage can also contain an auto-scale button 92 that, for example, appliesa standard offset and computes (and applies) a maximum amplification(minimum scaling) that results in a plot that is within the range of thedisplay. According to one embodiment of the present invention, when afile is initially loaded, plot 80 is auto-scaled.

Analyze-data page 70 can also include the capability to simultaneouslydisplay plots of the results from two separate measurements (e.g., twoseparate data files) for comparison purposes. For example, two data setscan be classified as primary and reference data. A separate group ofcontrols 94 for data file browse and load capability can be provided forthe reference file, while statistics and error diagnostics can bedisplayed for the primary data. A swap button 96 can also be provided toallow for exchanging the primary and reference files. Furthermore, abutton 98 to clear the reference file from the display can also beprovided.

Referring now to FIG. 4A, in an exemplary embodiment, analyze-data page70 contains a diagnostics button 100 that, when pressed, causes a new(e.g., diagnostics) window 102 to appear. Such a window 102 can contain,for example, a listing of error sources, their estimated values, and acheckbox for each error source. For example, checkboxes (e.g., checkbox104) can be used to indicate whether polar error plot 80 should containa contribution from the corresponding error source. In one embodiment ofthe present invention, the checkboxes are initially checked.Accordingly, unchecking such a checkbox (or several checkboxes) causesthe estimated effect(s) of the error(s) to be removed from themeasurement, and polar error plot 80 can be updated to reflect theadjusted measurement data. Diagnostics window 102 can also contain acheckbox 106 capable of enabling a user to select (or deselect) all theindividual error components with a single click.

The described capability can, for example, allow a user to predict thedegree to which applying compensation for one or more error types mightimprove the performance of the associated machine. One associatedadvantage could include determining which optional compensationcapabilities to activate in a control.

Referring now to FIG. 4, in a further exemplary embodiment, analyze-datapage 70 contains a cursor facility (such as one activated by pressingbutton 108) that aids a user in the task of manually assessing variousaspects of the radial errors. The capability can be used to allow theuser to place a polar cursor at any point on a trace on polar-error plot80. Displays that indicate the angle and the deviation from the meanradius for the cursor point can also be provided.

Such a cursor may be controlled by, for example, a slider bar thatincreases or decreases the angular location, entering values into anumeric field for circle angle, touching (or clicking) a location onplot 80, or by pressing arrow keys on a keypad to increase or decreasethe angular location. For example, when the cursor is moved by a slidercontrol or arrow keys, it may be moved at either a slow speed (where thecursor-jump interval is a single measurement point) or at a high speed(where the cursor jump interval is a fixed amount of degrees (e.g., 5degrees)).

Such a cursor facility might be useful when used in combination with adiagnostics capability for manually diagnosing cyclic errors that appearon, for example, ballbar plot 80. For example, cyclic errors can beisolated by using diagnostics window 102 to remove the effects of allthe estimated error types. The amplitude and period of the remainingerrors can then be manually identified with aid of a cursor.

Once again referring to FIG. 4A, in certain embodiments of the presentinvention, diagnostic window 102 can also be provided with the followingfunctionality: 1) the ability to choose the order in which errors aredisplayed (e.g., either ranked based on relative contribution to overallerror or grouped logically based on category), and/or 2) the ability tospecify whether a numeric error value display should indicate either anerror parameter or a contribution of that error to a total circularityerror. Diagnostics window 102 can also contain readouts 110 for ameasured circularity (e.g., a circularity error of the original(center-corrected) data), a corrected circularity (e.g., a circularityerror of adjusted data that has error sources of the types whosecheckboxes are checked), and/or a model coherence (e.g., a measurementof the ability of the error model to account for the content of themeasurement data).

Referring now to FIG. 5A, an adjust-compensation page 112 can also beincluded that provides an interface for a user to change configurationdata, such as that associated with, for example, certain types of axiscompensation that have an effect on a ballbar measurement. For example,such a page 112 can contain a pull-down menu 114 for selecting a majorplane (e.g., XY, XZ, YZ). Within such an adjust-compensation page 112,there can also be sub-pages associated with a particular type ofcompensation (e.g., six categories of compensation are described below).

In an exemplary embodiment of the present invention, within eachcategory of error, a configuration data display 116 is provided that islimited to the axes involved in the major-plane circular move. Theconfiguration data can be applied uniformly to all physical connectionsto the indicated logical axis (a detailed description of how cross-axisassociations can be created for multiple physical connections to alogical axis is further provided herein). In the descriptions, thehorizontal (H) and vertical (V) axes arc defined differently dependingon the major plane: {XY: H=X, V=Y}; {XZ: H=X, V=Z}; {YZ: H=Y, V=Z}). Thedisplays can be used to indicate which axis is horizontal and which isvertical.

A horizontal axis cross-comp sub-page 118 such as that depicted in FIG.5A can be included that contains configuration data to specify, forexample, a straightness and squareness compensation that compensates aV-axis position as a function of a H-axis position. In one embodiment,the specific logical axes that form the association are indicated on thepage header (e.g. Y=f(X)). Configuration items that can be on this pageinclude: 1) squareness coefficient, 2) straightness coefficient, 3)H-coordinate of compensation origin, and 4) straightness compensation'srange of applicability expressed in terms of ±displacement from thecompensation origin.

Similarly, a vertical axis cross-comp sub-page can be included thatcontains configuration data to specify a straightness and squarenesscompensation that compensates, for example, an H-axis position as afunction of the V-axis position. In an exemplary embodiment, theconfiguration items on this page mirror those on the horizontal axiscross-comp page discussed above and shown in FIG. 5A, except that theyapply to the V-axis rather than the H-axis.

Referring now to FIG. 5B, a horizontal axis scale-error-comp sub-page120 can be included that contains configuration data that is used tospecify, for example, pitch-error or scale-error compensation of theH-axis. For example, such a page can indicate which axis is the H-axisfor the particular major plane selected. The pitch or scale compensationcan be achieved by creating an association for squareness/straightnesscompensation where the axis is associated to itself, the squarenesscompensation coefficient is the pitch-error coefficient, and thestraightness compensation is disabled. In an exemplary embodiment, sucha page also contains fields for pitch-error-comp coefficient and acompensation origin for the H-axis.

Meanwhile, referring now to FIG. 5C, a vertical axis scale-error-compsub-page 122 can be included that contains configuration data that isused to specify scale-error-compensation for a V-axis. In an exemplaryembodiment, such a page mirrors that of the horizontalaxis-scale-error-comp page as described above, except that itcompensates for scale errors in the V-axis instead of the H-axis.

Referring now to FIG. 5D, a lost motion compensation sub-page 126 can befurther included that contains configuration data items for backlashcompensation and windup compensation for the H and V-axes involved inthe major plane circular move. In an exemplary embodiment, the specificlogical axes associated with the H and V axes are indicated in adisplay.

Furthermore, and referring now to FIG. 5E, a friction comp adjustmentssub-page 128 can be included that contains configuration data items forCoulomb-friction-feedforward parameters for the H and V-axes involved inthe major plane circular move, for example. In an exemplary embodiment,specific logical axes associated with the H and V axes are indicated ina display.

In further exemplary embodiments, adjust-compensation page 112 can alsocontain a commit button 130 that copies the values from the(current-values) numeric fields to, for example, a database associatedwith the control and then, for example, issues a configuration-changesystem event to update the configuration data.

Adjust-compensation page 112 can also provide suggested values forchanging existing compensation parameters. For example, eachconfiguration parameter can be represented by three fields on the page:an original value field 132, a current value field 134, and a proposedvalue field 136. In an exemplary embodiment, each configuration item'scurrent-value field is the source of that configuration item's valuethat is copied to a database when commit button 128 is pressed.

In one embodiment, the current-value field 134 is typically displayed.Meanwhile, in such an embodiment, the original-value and proposed-valuefields 132, 136 can be displayed when parameter estimates for a primaryfile have been obtained from the most recent measurement and when thecurrent database values for the compensation parameters are the same asthose present during the most recent test. One associated advantagecould be that this ensures that the compensation is adjusted using theappropriate data. One approach to accomplishing this involves displayingthe original-value and proposed-value fields 132, 136 when each of thefollowing two conditions are satisfied: 1) a primary file activelyloaded (e.g., on analyze-data page 70) has a file-reference-number thatmatches a data-reference-number (a unique numeric string constructed atthe beginning of test execution based on the test's major plane, date,and time) associated with the most recent test performed on the machine,and 2) the major plane selected on adjust-comp page 112 is the same asthe major plane of the primary file.

According to such an embodiment, the file-reference-number can beextracted from a data file header when the file is loaded. When the fileis originally created, the data-reference-number for the test can becopied to the file's header. The existing parametric compensation forthe machine plane can also be captured at the beginning of theexecution.

In an exemplary embodiment, when the currently loaded primary file isthe result of the most recent measurement, and the selected major planeon adjust-comp page 112 corresponds to the plane of the most recentmeasurement, then fields 132, 136 for original and proposed values foreach configuration item are displayed. In one embodiment, the numericcontents of these fields are not modifiable by the user. According tosuch an embodiment, the original value represents the value for thecompensation parameter that was active when the data was collected. Itcan be copied from the snapshot of configuration that was taken at thebeginning of test execution.

Meanwhile, the proposed value can be the value that was determined to bethe correct adjustment to compensation. The proposed value can becomputed based on the original value and the value for the associatederror parameter as estimated from the measurement data. In an exemplaryembodiment, adjust-comp page 112 provides the ability to copy thecontents of the proposed value fields into the current value fields, tocopy the contents of the original value fields to the current valuefields, and to clear and disable the compensation for the current valuefields. For example, buttons 138, 140, an 142 can be provided that arecapable of respectively initiating each of these tasks when selected. Inone such embodiment, each of these buttons could perform its operationsfor the fields on the current sub-page. Three additional buttons (e.g.,buttons 144, 146, and, 148) that perform these operations on allsub-pages can also be provided.

In addition to the development of the proposed values, the measured datacan also be used to determine a measure of statistical significance foreach of the proposed compensation parameters. As used herein,significance is a measure of whether an estimated error parameter is“significant” relative to the estimate for that error parameter'svariance. Mathematics behind the development of the significance measureare further discussed herein.

Referring back to FIG. 5D, the significance for each of the proposedvalues can be displayed graphically, such as by a meter 150. In such anembodiment, a value that causes a needle of meter 150 to be highindicates that the proposed change to the compensation parameter issignificant, while a value that causes the needle to be low indicatesthat the proposed change to the compensation parameter is insignificant.

In one embodiment of the present invention, the proposed value for acompensation parameter is determined by adding the (negative of the)latest estimated value for the error parameter to the compensationparameter that was active when the data was collected. Detaileddescriptions of how the proposed values for each compensation parameterare developed are further provided herein.

A detailed analysis of the processing involved in an exemplaryembodiment of the present invention involving ballbar testing is nowprovided. As part of the detailed description of the same by way ofexample, certain aspects of exemplary embodiments disclosed herein maybe specific to a particular commercially available ballbar (the QC10ballbar sold by Renishaw) and a particular motion control system (theA2100). One skilled in the art will, however, appreciate thatembodiments of the present invention can be applied to, for example,other devices (including other ballbars) and control systems. Inparticular, certain of the embodiments to be further described hereinrelate to application software currently called Ballbar Autotuning forParametric Compensation (ATB) and currently designed for use with theA2100 control, a detailed description of which will be provided hereinas an illustrative example.

As an introductory matter, the QC10 ballbar outputs its data to astandard RS232 serial port. Certain exemplary embodiments to bediscussed herein involve data collection and analysis running entirelywithin an operating system such as, for example, one currently soldunder one of Microsoft Windows trademarks. In some of these embodiments,the real time measurement data from the ballbar is acquired at aconstant sample rate and piped into memory accessible through theoperating system. In this way, a contiguous data set may be obtained ona non-real time operating system.

Thus, with such an approach, ballbar data is not exactly synchronizedto, for example, the real time internal data of the control. The type ofanalysis performed by certain embodiments of an exemplary embodiment asdisclosed herein does not require synchronization of the ballbar dataand internal control data. However, other embodiments of auto tuningmight involve such synchronization.

With respect to the analysis of measured data in an exemplary embodimentof the present invention, the raw measurement data can undergosignificant processing. For example, raw data can first be corrected foroffsets between the center of rotation of the ballbar and the center ofthe circle. The circularity-error (a measure of the combined effect ofall errors) can be extracted from the center-corrected data.

Next, estimates for the error parameters can be extracted from the data.During the parameter estimation, the coherence of the data can bedetermined. Once estimates for error parameters are obtained, individualeffects of each error can be determined and the effect of the error maybe removed from the data in response to, for example, user manipulationof the checkboxes of diagnostics window 102. Similarly, the relativeimpact of each error type on the overall circularity may be assessed.

Much of the data analysis involved in this specific example employs“curve-fitting” methods to extract parameters from the raw data. As usedherein, the term “curve fitting” describes a process where the measureddata is used to determine the parameters for a mathematical model of anassumed structure or framework. Ideally, curve fitting determines thevalues for the set of parameters in the mathematical model that have themaximum likelihood of having created the measured data.

For the parametric machine errors detected by an exemplary embodiment ofthe present invention, the curve fitting process involves aleast-squares optimization. A least squares estimator can be a maximumlikelihood estimator if each measurement error is normally distributedand each measurement error has the same variance. This might be untruefor the present case, but the least squares estimator can be used anywaybecause it tends to provide a decent result and there is no readilyavailable alternative given the available a-priori information about thedata. A relatively straightforward linear least-squares approach may beemployed because the assumed structure of the radial error involves alinear combination in the model parameters.

A linear least squares formulation can be based on an overdeterminedsystem where a linear combination of basis functions scaled by the modelparameters yields the measurement. In this specific case, the basisfunctions can be functions of the (constant) nominal circle radius andthe (varying) circle angle. The measurement can be a function of theradius measurements. The formulation may be written in matrix form asfollows, where {P} is the m-element parameter vector, {f(r(t))} is the nelement vector where each element is a function of measured ballbarradius at time t, and [A(R,θ(t))] is the n×m matrix whose columns arebasis functions developed from the circle angle and nominal circleradius.

 {fr(t)}=[A(R,θ(t))]{P}  (1)

If n>m, Equation (1) represents an overdetermined system where there aremore knowns than unknowns. This equation can be solved using the leastsquares technique which minimizes the sum of the squares of the errorsbetween the modeled f(r) and the measured f(r).Σe ² ={[A]{P}−f(r)}^(T) ·{[A]{P}−f(r)}  (2)The solution for the parameter vector {P} that minimizes this quantityincludes the least squares solution and can be written using theso-called normal equations.{P}=[A] ^(T) ·[A] ⁻¹ ·[A] ^(T) ·{f(r)}  (3)The solution involves inversion of the matrix [[A]^(T)[A]]. This matrixhas the properties of being real, symmetric, and positive-definite.These properties can be taken advantage of when finding a technique forinverting the matrix. The matrix inverse can be computed using Jacobi'smethod for finding the eigenvalues and eigenvectors of a real symmetricpositive definite matrix.

Briefly stated, Jacobi's method involves successively applying rotationtransformations to portions of the matrix until it is diagonalized. Afull explanation of Jacobi's method is given in “Numerical Recipes in C”by Press, Flannery, Teukolsy and Vettering, which is hereby incorporatedherein by reference. The Jacobi technique should work as long as thematrix is invertible. Another advantage of using Jacobi's technique forsolving the normal equations is that it provides the eigenvalues of[A]^(T)[A]. These eigenvalues can be used to determine the conditionnumber of the matrix [A]—a factor related to the excitation provided bythe axis motion.

The topics that follow related to the development of the equations thatdescribe the effects of each parametric machine error on a ballbarradius measurement. Each of these equations can be combined into anindividual overdetermined system formulation in such a way that theerror parameters are estimated simultaneously. This approach minimizespossibility of incorrectly attributing measurement aspects to the wrongerror type.

The equations can be developed in such a way to enable a description ofthe effect of each error type in terms of the measured ballbar radius.The following equations are developed for the case of the X-Y plane.These results are equally applicable to the XZ and YZ planes. The x andy designators used in this development may be interpreted to mean thehorizontal and vertical axes of the circle-plane rather than the machinetool axes.

It is often the case that the circle center is not coincident with thecenter ball of the ballbar base fixture. These differences can occur ifthe center ball shifts when the set screw is tightened, if the servooffsets drift with time, the axis has a significant straightness error(discussed later), or any number of other reasons. In order to properlyanalyze the measurement, the effect of center offset can be consideredalong with other parametric error effects.

In an exemplary embodiment, center offset is not considered to be amachine error; it is considered to be an error associated withmeasurement. The formulation of the least squares equations fordetermining center offset can be developed by considering the equationsfor the length of the vector from the centering fixture location to apoint on the ideal circle as a function of circle angle. The situationcan be depicted graphically as shown in FIG. 6A. A polar error plot forthe ballbar measurement with a center offset error contains a “kidneybean” signature of the type shown in FIG. 6B.

Examination of FIG. 6A leads to the development of the followingequation for the measured radius r as a function of the circle angle θ,where R is the average circle radius that may differ from the programmedradius if the axes have scaling errors (discussed later).R cos(θ)−E _(CTR) _(—) _(X) ² +R sin(θ)−E _(CTR) _(—) _(Y) ² =r(θ)²  (4)The equation can be expanded to the following form.R ²−2RE _(CTR) _(—) _(X)cosθ−2RE _(CTR) _(—) _(Y)sinθ+E _(CTR) _(—) _(X)²+E _(CTR) _(—Y) ²=r(θ)²  (5)For the case of multiple measurements, the previous equation can berepresented in the matrix form to accommodate the least squaressolution. $\begin{matrix}{{\begin{bmatrix}{{- 2}{\cos\left( {\theta\left( t_{1} \right)} \right)}} & {{- 2}{\sin\left( {\theta\left( t_{1} \right)} \right)}} & 1 \\{{- 2}{\cos\left( {\theta\left( t_{2} \right)} \right)}} & {{- 2}{\sin\left( {\theta\left( t_{2} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{{- 2}{\cos\left( {\theta\left( t_{n} \right)} \right)}} & {{- 2}{\sin\left( {\theta\left( t_{n} \right)} \right)}} & 1\end{bmatrix}\begin{Bmatrix}{RE}_{CTR\_ X} \\{RE}_{CTR\_ Y} \\{R^{2} + E_{CTR\_ X}^{2} + E_{CTR\_ Y}^{2}}\end{Bmatrix}} = \begin{pmatrix}\left( {r\left( t_{1} \right)} \right)^{2} \\\left( {r\left( t_{2} \right)} \right)^{2} \\\vdots \\\left( {r\left( t_{n} \right)} \right)^{2}\end{pmatrix}} & (6)\end{matrix}$The previous equation may be “solved” using equation (3) to obtain theparameter vector where the matrix to the left is [A], the 3×1 vector isP and the n×1 vector is f(r). The equation used to determine the circlecenter offsets also provides an average circle radius R. Three desiredparameters, R, E_(CTR) _(—) _(X), and E_(CTR) _(—) _(Y), may beextracted from the vector P by an iterative technique.

A first approximation of setting the quantity E_(CTR) _(—) _(X)²+E_(CTR) _(—) _(Y) ² to be zero allows the parameters for the firstiteration to be extracted directly. Typically, the center offset is verysmall relative to the circle radius and iterations are not needed.However, for the sake of robustness, an iterative algorithm can be madeavailable in case it is needed. The iterative algorithm described belowconverges to the correct result as long as the ballbar center is insidethe circle.

P = [A^(T)A]⁻¹A^(T)r² R_(est) =(P₃)^(0.5,)R_(est) _(—) _(prev) = 0E_(CTR) _(—) _(X) = P₁/R_(est,) E_(CTR) _(—) _(Y) = P₂/R_(est) DoForever R_(est) = (P₃ − (E_(CTR) _(—) _(X))² − (E_(CTR) _(—)_(Y))²)^(0.5) If |R_(est) -R_(est) _(—) _(prev)| > 0.01 μ, Then, BreakE_(CTR) _(—) _(X) = P₁/R_(est,) E_(CTR) _(—) _(Y) = P₂/R_(est) R_(est)_(—) _(prev)=R_(est); Continue

Once the values for the circle center offsets are known, they can beused in manipulation of the raw ballbar radius data r_(raw) to removethe effects of center offsets from the data. The center offset effectscan be removed when computing the circularity statistic. An approximateequation for removing the effects of the center offsets can be obtainedby examination of FIG. 6 and making the assumption that the angle of theballbar relative to the horizontal axis is the same as the circle angle.$\begin{matrix}{{r_{ctr\_ adj}(\theta)} = \sqrt{\left( {{{r_{raw}(\theta)} \cdot {\cos(\theta)}} + E_{CTR\_ X}} \right)^{2} + \left( {{{r_{raw}(\theta)} \cdot {\sin(\theta)}} + E_{CTR\_ Y}} \right)^{2}}} & (7)\end{matrix}$An equation for adjusting the measured radius for center offsets can beobtained by examining FIG. 6 and employing the law of cosines where thex_(ce) and y_(ce) designate the estimates for center offsets E_(CTR)_(—) _(X), E_(CTR) _(—) _(Y). $\begin{matrix}{\left\lbrack {r_{ctr\_ adj}(\theta)} \right\rbrack^{2} - {\left( {2\sqrt{x_{ce}^{2} + y_{ce}^{2}}{\cos\left( {{\tan^{- 1}\left( {y_{ce},x_{ce}} \right)} - \theta} \right)}} \right) \cdot \left\lbrack {r_{ctr\_ adj}(\theta)} \right\rbrack} + \left( {x_{ce}^{2} + y_{ce}^{2} - {r_{raw}^{2}(\theta)}} \right)} & (8)\end{matrix}$

A solution for the center-adjusted radius is the positive root of theprevious quadratic equation. The roots should be real as long as thecenter is inside of the circle. The circularity-error or circularity isa measure of the maximum deviation of the out-of-roundness of thecircle. It can be defined as the maximum center-adjusted radius minusthe minimum center adjusted radius.circularity=maxr _(ctr) _(—) _(adj)(θ)−minr _(ctr) _(—) _(adj)(θ)  (9)In an exemplary embodiment, the computed circularity from themeasurement is displayed. The circularity for data that has beenmathematically manipulated in such a way that the effects of acombination of user-selected parametric errors can be removed can alsobe computed and/or displayed.

The most common machine tool design involves an arrangement where linearmotion within a plane is effected by coordinating the motion of twoindividual linear displacement actuator mechanisms (axes) that areperpendicular to each other, i.e., the axes are square. The machine toolmanufacturing process may result in a squareness error where the axesare not precisely perpendicular to each other as is intended. Thisdeviation from perpendicularity is known as the squareness error orsimply squareness.

The squareness error describes a relationship between the two axes inthe plane with one squareness error per major plane. Individual proposedvalues for squareness compensation for each axis in the plane can alsobe developed by equally distributing the total squareness compensationamong the two axes. The equation for the effect of a squareness error onthe circle radius can be developed based on FIG. 7.

The coordinates of a point on the actual (non-circular) curve as afunction of circle angle θ can be described by the following equationswhere R is the average radius which may differ from the programmedradius if the machine also has a scale error.x(θ)=R cos(θ)+R sin(θ)sin(½φ)−R cos(θ)sin(½φ)tan(½φ)  (10)y(θ)=R sin(θ)+R cos(θ)sin(½φ)−R sin(θ)sin(½φ)tan(½φ)  (11)In these equations, the final terms on the right that involve thequantity sin(½φ)tan(½φ) are insignificantly small for small values of φ.This is a valid assumption for the amplitude of squareness errors foundon most machine tools. The ballbar radius as a function of circle angleis the desired result and can be developed by computing the sum of thesquares of the x and y components of the point on the curve andemploying the aforementioned simplifying assumption.r ²(θ)=x ²(θ)+y ²(θ)  (12)r ²(θ)=R ²cos²(θ)+sin²(θ)1+sin²(½φ)+2R ² sin(θ)cos(θ)sin(½φ)  (13)For small φ, the term sin²(½φ) becomes insignificant and the previousequation may be simplified to the followingr ²(θ)=R ²1+2 sin(θ)cos(θ)sin(½φ)  (14)The parameter that describes the squareness error in the X-Y plane canbe defined as the unitless proportionality factor sin(½φ). Analogousparameters can be obtained for the X-Z and Y-Z planes.E _(SQR) _(—) _(XY)=sin(½φ)  (15)The following matrix equation is for an overdetermined system ofequations represented as a linear combination of terms in the squarenessparameter. $\begin{matrix}{\begin{Bmatrix}\left( {r\left( t_{1} \right)} \right)^{2} \\\left( {r\left( t_{2} \right)} \right)^{2} \\\vdots \\\left( {r\left( t_{n} \right)} \right)^{2}\end{Bmatrix} = {\begin{bmatrix}{2{\sin\left( {\theta\left( t_{1} \right)} \right)}{\cos\left( {\theta\left( t_{1} \right)} \right)}} & 1 \\{2{\sin\left( {\theta\left( t_{2} \right)} \right)}{\cos\left( {\theta\left( t_{2} \right)} \right)}} & 1 \\\vdots & \vdots \\{2{\sin\left( {\theta\left( t_{n} \right)} \right)}{\cos\left( {\theta\left( t_{n} \right)} \right)}} & 1\end{bmatrix}\begin{Bmatrix}{R^{2}E_{SQR}} \\R^{2}\end{Bmatrix}}} & (16)\end{matrix}$

The axis squareness error is most often described as the perpendiculardeviation in units of parts-per-million or equivalently microns/meter orinches/mil. This is the way the squareness compensation is representedand the way the squareness error can be displayed, such as on a screen.The conversion from the unitless squareness parameter toparts-per-million squareness can be obtained by multiplying the unitlessparameter by 1 million.squareness=1,000,000·E _(SQR)(μ/m)  (17)

A conventional linear motion machine tool axis includes an accuratefeedback measurement to move to the correct position. Axis scale errorscan be created by an inaccurate feedback mechanism or an inaccurateparameter for converting a feedback measurement from, for example, arotary device to linear displacement. One common linear axis mechanisminvolves a rotary motor that drives a ballscrew. A ball-nut converts therotary motion to linear motion. The axis position may be estimated bymeasuring the motor angle and computing the axis position based on anassumed value for screw pitch. Alternatively, the axis position may bemeasured using a linear displacement sensor such as a glass scale, forexample.

In the first case, differences between the assumed and actual pitch ofthe ballscrew can cause the axis to travel an incorrect distance due tothis pitch error. In the second case, errors in the scale itself, due tofactors such as thermal growth, for example, can cause the axis totravel to an incorrect position due to the scale error. Pitch error andscale error can cause an axis to travel a distance that is differentthan desired.

This type of error also appears on other axis mechanisms, such asrack-and-pinion and linear motor driven axes. The fundamental type oferror experienced with these types of mechanisms is a position errorthat is linearly proportional to the distance traveled. The constant ofproportionality may be represented using units such as parts-per-million(equivalent to microns/meter or mils/inch).

Effects of axis linear-scale-errors on circular motion in a plane areshown in FIG. 8. The figure shows the actual circle (not a polar errorplot) for a case where the scale errors are very large relative to theradius. The figure shows how the two scale errors from each axis acttogether to create two effects: 1) a relative scale mismatch between theaxes that causes the circle to be stretched or compressed in a directionparallel to the axis, and 2) an overall enlargement or reduction in theaverage (best-fit) radius of the (distorted) circle. The specificexample shown in FIG. 8 is for a case where the scale error on the Xaxis is positive (the axis moves further than desired) and the scaleerror on the Y axis is negative (the axis moves a smaller distance thanis desired). Since the (positive) X axis scale error is larger than the(negative) Y axis error, the average circle radius (best-fit-circle) islarger than the ideal circle radius.

The equations for the effect of each scaling error on the measuredcircle radius can be developed by first considering the x and y axesseparately.x(θ)=R _(prog) cos(θ)1+E _(SCL) _(—) _(X) , y(θ)=R _(prog) sin(θ)1+E_(SCL) _(—) _(Y)  (18)The square of the instantaneous radius is the sum of the squares of thex and y vector components. With the (valid) assumption that the squareof the error parameters are insignificant, the following relation can beobtained. $\begin{matrix}{{r^{2}(\theta)} = {R_{prog}^{2} + {2R_{prog}^{2}E_{SCL\_ X}{\cos^{2}(\theta)}} + {2R_{prog}^{2}E_{SCL\_ Y}{\sin^{2}(\theta)}}}} & (19)\end{matrix}$The above equation can be reconstructed as a n^(x)2 matrix equation.This can be done if the ballbar is length calibrated such that thedifference between r²(θ) and R_(prog) is meaningful.

In an exemplary embodiment, differences between x and y scaling errorsare accounted for even if the ballbar is not length-calibrated. Oneapproach involves transforming the individual X and Y scale errors intoa scale mismatch error and an average radius. If the ballbar is notlength-calibrated, then the estimate for average radius can bediscarded. If the ballbar is length-calibrated, then the estimate foraverage radius can be used to extract the individual axis scaling errorsfrom the scale mismatch error. The parameters in the new formulationhave the following definitions: $\begin{matrix}{{E_{SCL\_ XY} = \frac{E_{SCL\_ X} - E_{SCL\_ Y}}{2}},{R_{avg} = {R_{prog}\left( {1 + \frac{E_{SCL\_ X} + E_{SCL\_ Y}}{2}} \right)}}} & (20)\end{matrix}$

If the scaling error is small enough to make the square of the errorterms insignificant, then the following equations for the x and ylocations on the curve may be used. These equations assume that thecircle is stretched in the x-direction; if the circle is actuallystretched in the y-direction, then the scale mismatch parameterE_(Smismatch) will be a negative number.x(θ)=R _(avg) cos(θ)1+E _(SCL) _(—) _(XY) , y(θ)=R _(avg) sin(θ)1−E_(SCL) _(XY)  (21)The sum of each component squared in the previous equation yields theexpression for the instantaneous ballbar radius (where the error termsquared can be once again ignored). $\begin{matrix}{{r^{2}(\theta)} = {R_{avg}^{2} + {2{R_{avg}^{2}\left( {{\cos^{2}(\theta)} - {\sin^{2}(\theta)}} \right)}E_{SCL\_ XY}}}} & (22)\end{matrix}$The previous equation can be represented in the following matrix formfor measurements at corresponding times. $\begin{matrix}{\begin{Bmatrix}\left( {r\left( t_{1} \right)} \right)^{2} \\\left( {r\left( t_{2} \right)} \right)^{2} \\\vdots \\\left( {r\left( t_{n} \right)} \right)^{2}\end{Bmatrix} = {\begin{bmatrix}{2\left( {{\cos^{2}\left( {\theta\left( t_{1} \right)} \right)} - {\sin^{2}\left( {\theta\left( t_{1} \right)} \right)}} \right)} & 1 \\{2\left( {{\cos^{2}\left( {\theta\left( t_{2} \right)} \right)} - {\sin^{2}\left( {\theta\left( t_{2} \right)} \right)}} \right)} & 1 \\\vdots & \vdots \\{2\left( {{\cos^{2}\left( {\theta\left( t_{n} \right)} \right)} - {\sin^{2}\left( {\theta\left( t_{n} \right)} \right)}} \right)} & 1\end{bmatrix}\begin{Bmatrix}{R_{avg}^{2}E_{SCL\_ XY}} \\R_{avg}^{2}\end{Bmatrix}}} & (23)\end{matrix}$

If the ballbar is not length calibrated, then the estimate for R_(avg)is meaningless and the individual axis scale errors cannot be extractedfrom the scale mismatch errors. In this case, values for the individualaxes' scale errors can be computed by assuming that each axiscontributes equally toward the observed scale mismatch.

 E _(XCL) _(—) _(X) =E _(SCL) _(—) _(XY) , E _(SCL) _(—) _(Y) =−E _(SCL)_(—) _(XY)  (24)

If the ballbar is length calibrated, then the individual axis scaleerrors may be extracted using the following equations. $\begin{matrix}{{E_{SCL\_ X} = {\frac{R_{avg}}{R_{prog}} - 1 + E_{SCL\_ XY}}},{E_{SCL\_ Y} = {\frac{R_{avg}}{R_{prog}} - 1 - E_{SCL\_ XY}}}} & (25)\end{matrix}$The individual axis scale errors and the scale mismatch errors in theprevious equation are unitless factors of proportionality. It is common,however, to represent these factors in parts-per-million units, alsodescribed as microns/meter or mils/inch. The conversion of a scale error(E_(SCL) _(—) _(X), E_(SCL) _(—) _(Y), E_(SCL) _(—) _(XY)) from aunitless factor to a parts-per-million factor can be simply a matter ofmultiplying the scale error term by a factor of 10⁶.

In the preceding discussions, it has been assumed that an axis' linearscale error is the phenomenon that can cause the average circle radiusto differ from the ideal radius. The other parametric error typesconsidered do not appear to affect the average circle radius.Specifically, the average circle radius does not appear to be affectedby squareness errors, straightness errors, lost-motion errors, and servomismatch errors.

There is, however, one phenomenon in addition to scale errors that doesappear to affect the average circle radius. The dynamic characteristicsof a position controller can result in amplitude attenuation (oramplification) that varies according to circle radius and feedrate.Position controllers have a low-pass frequency characteristic, and somemay exhibit amplification at certain frequencies. Since the circlefrequency is the per-second-feedrate divided by the radius, thefrequency increases with increases in feedrate or reductions in nominalradius. For cases where the ballbar is length calibrated, an adjustmentcan be made to the estimate of average radius to account for theexpected behavior of the position loop's dynamic response.

The expected behavior of the position loop response can be based on anumber of approximating assumptions. The assumptions are expected to bereasonable for circle frequencies that are below the cutoff frequency ofthe position loop. Even the highest feedrate ballbar tests are expectedto correspond to frequencies that are within the bandwidth of the axesfor even the lowest gain controllers. For example, an axis with a Kvgain of 1.0 and no feedforward has a cutoff frequency of approximately2.7 Hertz; for a circle radius equal to the minimum ballbar length of100 mm, the feedrate associated with the cutoff frequency is 100,000mm/min—a value beyond the capability of most machines and higher thanwhat would typically be used during a ballbar test. The assumptionsabout the dynamic response of the axes can be as follows:

-   -   1. Both axes that actuate motion in the circle plane have        identical values for Kv and VFF.    -   2. Both axes have identical amplitude response up to the cutoff        frequency.    -   3. The effective Kv is identical to the nominal Kv adjusted by        the Kv_(adjust).    -   4. The axis dynamics are approximated as a continuous time        system with an infinite bandwidth velocity loop.

Based on these assumptions, the effect of axis dynamic response on theaverage radius can be developed from the system shown in FIG. 9. Atransfer function of this simplified model for a position controller isshown below. $\begin{matrix}{{\frac{x_{fdbk}}{x_{cmd}}(s)} = \frac{{\left( {{VFF}/100} \right)s} + \left( {1000{{Kv}/60}} \right)}{s + \left( {1000K\quad{v/60}} \right)}} & (26)\end{matrix}$The amplitude scaling for a sinusoidal input of angular frequency ω canbe determined by evaluating the magnitude of the transfer function ats=jω. $\begin{matrix}{{{\frac{x_{fdbk}}{x_{cmd}}\left( {s = {j\omega}} \right)}}\sqrt{\frac{{\left( {{VFF}_{X}/100} \right)^{2}\omega^{2}} + \left( {1000{{Kv}_{X}/60}} \right)^{2}}{\omega^{2} + \left( {1000{{Kv}_{X}/60}} \right)^{2}}}} & (27)\end{matrix}$

For a constant feedrate circular move, the axis motion can be sinusoidalwhere ω=Feedrate/(60R). Since both axes are assumed to have identicalresponses, the radius amplitude scaling is identical to the axisamplitude scaling. Performing the substitution for ω, yields the finalresult for the radius scaling due to the position loop dynamic response.$\begin{matrix}{R_{{avg}\quad{dy}\quad n} = {R_{prog} \cdot \sqrt{\frac{{\left( {{VFF}/100} \right)^{2}\left( {{Feedrate}/R_{prog}} \right)^{2}} + \left( {1000{{Kv}/60}} \right)^{2}}{\left( {{Feedrate}/R_{prog}} \right)^{2} + \left( {1000{{Kv}/60}} \right)^{2}}}}} & (28)\end{matrix}$

The dynamic response can be factored into the estimate of the axis scaleerrors by measuring the overall scaling relative to the expected radius,given the programmed circle radius and the model for dynamic response.$\begin{matrix}{E_{SCL\_ X} = {{\frac{R_{avg}}{R_{prog}}\sqrt{\frac{\left( {{Feedrate}/R_{prog}} \right)^{2} + \left( {1000{Kv}} \right)^{2}}{{\left( {{VFF}/100} \right)^{2}\left( {{Feedrate}/R_{prog}} \right)^{2}} + \left( {1000{Kv}} \right)^{2}}}} - 1 + E_{SCL\_ XY}}} & (29) \\{E_{SCL\_ Y} = {{\frac{R_{avg}}{R_{prog}}\sqrt{\frac{\left( {{Feedrate}/R_{prog}} \right)^{2} + \left( {1000{Kv}} \right)^{2}}{{\left( {{VFF}/100} \right)^{2}\left( {{Feedrate}/R_{prog}} \right)^{2}} + \left( {1000{Kv}} \right)^{2}}}} - 1 - E_{SCL\_ XY}}} & (30)\end{matrix}$

An axis straightness error is a linear displacement error in a directionperpendicular to the axis containing the straightness error.Straightness errors can be treated in a similar way as squareness errorssuch that the linear displacement errors that have a direction that isparallel to the direction of one of the machine's linear axes (e.g., x,y, and z) are considered. In other words, the straightness errors can beconsidered for one machine major-plane at a time.

Straightness errors can be distinguished from squareness errors in thatthe relationship between the error in the direction of theperpendicular-axis relative to the position of the axis containing theerror is non-linear. In the present discussion, the non-linearrelationship is restricted to the case of a symmetric parabolic curvewhere the axis of symmetry passes through the location of the center ofthe circle in the ballbar test. The assumed structure for thestraightness error of the X axis in the direction of the Y axis can bedefined by the following equation, where the deviation caused bystraightness errors is ΔY, X₀ is the x coordinate of the center of thetest circle, and E_(STR) _(—) _(X) _(—) _(Y) is the X-axis straightnesserror factor for the Y directed component of straightness error.

 Δy _(STR)(x)=E _(STR) _(—) _(X) _(—) _(Y)(x−x ₀)²  (31)

The assumption that a straightness error is reasonably approximated by aparabolic curve can be based on experience with machine tools that hasshown that axis guideways often have a fundamental straightness errorthat involves a bowing shape. Limiting the straightness error to includethe fundamental component prevents the analysis from accounting forerrors where the guideway has multiple bows or wiggles. Straightnesserrors that are more complex than a fundamental component can be handledusing a data-based approach, such as developing tables based on lasermeasurements and employing the cross-axis error compensation.

The assumption that the parabola is symmetric about a line parallel tothe perpendicular axis can be a reasonable one because the squarenesserror should account for the presence of any linear trend. Theassumption that the parabola's axis of symmetry passes though the centerof the test circle x₀ is not based on any physical insight about themachine, since the location of the circle center is arbitrary and maychange from test to test. This restriction can be included because itgreatly simplifies the analysis, enabling the error parameters to beestimated from a single evaluation of the least-squares normalequations. This restriction does not limit the ability to identify errorsources within the range of the ballbar test because an expansion ofequation (31), shown below in equation (32), reveals that the choice ofx₀ may be arbitrary since it is a factor in: 1) the linear trendaccounted for by the estimate for squareness error and 2) a y-componentof center offset accounted for by the estimate of center offset error.E _(STR) _(—) _(X) _(—) _(Y)(x−x ₀)²=(E _(STR) _(—) _(X) _(—) _(Y))·x²−(2E _(STR) _(—) _(X) _(—) _(Y) x ₀)·x+(E _(STR) _(—) _(X) _(—) _(Y) x₀ ²)  (32)

The choice for x₀ may be arbitrary in the sense that the least squaresapproach should provide an optimal result by providing an estimate forsquareness and y-center offset that accommodates the particular choiceof x₀ for straightness. However, the arbitrary choice for x₀ makesextrapolation of the error model outside the range of the test invalid.Application of compensation based on the estimate to locations outsideof the range of the test can easily cause larger errors to occur inthese regions than those that might have occurred without anycompensation.

The presence of the linear trend constant terms in equation (32) impliesestimating the squareness error and center offset errors at the sametime as the straightness error. Accordingly, in an exemplary embodiment,the center correction is not applied to the raw data prior to performingthe diagnostics. This can be potentially important for data arcs thatcover less than 360 degrees.

An example of a situation in the X-Y plane where the Y-axis has nostraightness-error and the X-axis has a straightness error that complieswith the assumptions is shown in FIG. 10. The plot shows how astraightness error might affect the shape of a curve on a polar errorplot with a high degree of amplification. The equation for the distortedcurve can be developed by applying the error from equation (31) to theequation for a circle where R is the actual best-fit circle radius thatmay differ from the nominal radius due to a scale error.x=R·cos(θ), y=R·sin(θ)+E _(STR) _(—) _(X) _(—) _(Y) ·R·cos(θ)²  (33)

The square of the radius is the sum of the squares of the x and ycoordinates of the curve. If terms involving the square of thestraightness coefficient are ignored by assuming that the straightnesserror is small, then the following equation for circle radius as afunction of angle can be derived.r ²(θ)=R ²+2E _(STR) _(—) _(X) _(—) _(Y) R ³sin(θ)cos²(θ)  (34)An analogous equation for the impact of a Y-axis straightness error onthe instantaneous circle radius may be derived in a similar manner. Anequation for the effect of the Y-axis straightness error E_(STR) _(—)_(Y) _(—) _(X) is shown below.r ²(θ)=R ²+2E _(STR) _(—) _(Y) _(—) _(X) R ³cos(θ)sin²(θ)  (35)The combined effect of both planar straightness errors is shown belowwith the equations written for multiple measurements in matrix form.$\begin{matrix}{\begin{Bmatrix}{r^{2}\left( t_{1} \right)} \\{r^{2}\left( t_{2} \right)} \\\vdots \\{r^{2}\left( t_{n} \right)}\end{Bmatrix} = {\quad{\begin{bmatrix}{2{\sin\left( {\theta\left( t_{1} \right)} \right)}{\cos^{2}\left( {\theta\left( t_{1} \right)} \right)}} & {2{\cos\left( {\theta\left( t_{1} \right)} \right)}{\sin^{2}\left( {\theta\left( t_{1} \right)} \right)}} & 1 \\{2{\sin\left( {\theta\left( t_{2} \right)} \right)}{\cos^{2}\left( {\theta\left( t_{2} \right)} \right)}} & {2{\cos\left( {\theta\left( t_{2} \right)} \right)}{\sin^{2}\left( {\theta\left( t_{2} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{2{\sin\left( {\theta\left( t_{n} \right)} \right)}{\cos^{2}\left( {\theta\left( t_{n} \right)} \right)}} & {2{\cos\left( {\theta\left( t_{n} \right)} \right)}{\sin^{2}\left( {\theta\left( t_{n} \right)} \right)}} & 1\end{bmatrix} \cdot \begin{Bmatrix}{R^{3}E_{{STR\_ X}{\_ Y}}} \\{R^{3}E_{{STR\_ Y}{\_ X}}} \\R^{2}\end{Bmatrix}}}} & (36)\end{matrix}$

As mentioned previously, the estimation of straightness errors can beperformed simultaneously with the estimation of squareness and centeroffset; thus, in an exemplary embodiment, the previous matrix equationis not used directly. A global matrix equation that simultaneouslyestimates the parameters is further provided herein.

A servo mismatch error is an error created due to a difference in thedynamic response between the two axes of the circle plane. The dominanteffect of servo mismatch is a phase mismatch. A previous discussionherein described how the amplitude response of an axis is expected to beflat beyond the highest conceivable circle frequency, even for low Kvgains. This means that the Kv gains may be mismatched without creating amismatch in the amplitude response.

However, this is not the case for the phase response which, for aposition loop without feedforward, begins to roll off immediately. Forany given circle-frequency, i.e., feedrate/radius, the phase differencebetween the sinusoids for the x and y axes is desired to be 90 degrees,but may differ from this ideal amount by some constant deviation. Theconstant deviation should vary with circle frequency, and, in mostcases, the deviation is more likely to be significant as the circlefrequency is increased (but the relationship is not necessarilymonotonic).

The phase deviation can be caused by mismatches in the effectiveposition loop gain, velocity loop response, or both. It is notconventionally possible to diagnose the source of the phase mismatchwith a single ballbar measurement—which can be accomplished byperforming frequency response measurements for each axis individually.Even though no attempt to propose a compensation for an observed servomismatch error is made, a servo-mismatch error can be computed becauseit informs the user that such an error is present, prevents servomismatch errors from being falsely attributed to other error types, andallows the user to gauge the servo-mismatch error's impact on thecircularity and to predict the improvement that might be accomplished ifthe error were removed.

The combined phase error between the two axes during a constant feedratecircle may be represented by the following equation where the totalphase error φ is equally divided among the two axes by assuming a half φlead on the X axis and a half φ lag on the Y-axis. For acounter-clockwise move, the circle angle θ increases with time (i.e.,the frequency dθ/dt=ω is positive). For clockwise moves, the reverse istrue—the frequency is negative. Equations for these two cases can beshown below.X _(CCW)(t)=R·cosωt+½φ|ω|, Y _(CCW)(t)=R·sinωt−½φ|ω|  (37)X _(CLW)(t)=R·cos−ωt+½φ|ω|, Y _(CLW)(t)=R·sin−ωt−½φ|ω|  (38)

In these equations, ω is the per-second feedrate divided by theprogrammed radius and, as mentioned, φ is expected to be constant for agiven fixed value of |ω|. The analysis will proceed first for thecounter-clockwise (positive-frequency) case. The results for theclockwise case can be derived in an analogous manner. The square of theradius for the counter-clockwise case is the sum of the squares of theaxis locations. In the following equation, the term ωt is replaced bythe circle angle θ. $\begin{matrix}{{r_{CCW}^{2}(\theta)} = {R^{2} \cdot \left( {{\cos^{2}\left( {\theta + {\frac{1}{2}{\phi\left( {\omega } \right)}}} \right)} + {\sin^{2}\left( {\theta - {\frac{1}{2}{\phi\left( {\omega } \right)}}} \right)}} \right)}} & (39)\end{matrix}$The previous equation can be simplified by employing some trigonometricidentities to the following form. $\begin{matrix}{{r_{CCW}^{2}(\theta)} = {R^{2} + {{R^{2} \cdot 2}\quad{\sin(\theta)}{\cos(\theta)}{\sin(\phi)}}}} & (40)\end{matrix}$

Now returning to the clockwise case, the equation (38) can bereformulated by changing the sign of the arguments of the sine andcosine functions.X _(CLW)(t)=R·cosωt−½φ|ω|, Y _(CLW)(t)=−R·sinωt+½φ|ω|  (41)These equations can be used to obtain the following result for theinstantaneous radius of the clockwise circle. $\begin{matrix}{{r_{CLW}^{2}(\theta)} = {R^{2} - {{R^{2} \cdot 2}\quad{\sin(\theta)}{\cos(\theta)}{\sin(\phi)}}}} & (42)\end{matrix}$The impact of a servo mismatch error on a polar error plot is shown inFIG. 11.

Comparison of equations (40) and (42) to equation (14) indicates thatthe formulation for servo mismatch error can be the same as theformulation for squareness error. This means that the two errors can beindistinguishable unless data for both clockwise and counterclockwisemeasurements are present in the matrix formulation. Thus, in anexemplary ballbar test, each measurement contains both a clockwise andcounter-clockwise run, and for the sake of equal weighting, both runscover the same data-range. This can be accomplished by “stacking” thematrix with rows developed from the counter-clockwise measurement on topof rows developed from the clockwise measurement. In the followingequation, the superscripts clw and ccw on the i'th measurement indicatorti are used to identify whether a measurement is from a clockwise orcounter-clockwise test. $\begin{matrix}{\begin{Bmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{r^{2}\left( t_{1}^{ccw} \right)} \\{r^{2}\left( t_{2}^{ccw} \right)}\end{matrix} \\\vdots\end{matrix} \\{r^{2}\left( t_{n}^{ccw} \right)}\end{matrix} \\\ldots\end{matrix} \\{r^{2}\left( t_{1}^{clw} \right)}\end{matrix} \\{r^{2}\left( t_{2}^{clw} \right)}\end{matrix} \\\vdots\end{matrix} \\{r^{2}\left( t_{n}^{clw} \right)}\end{Bmatrix} = {\begin{bmatrix}{2\sin\quad{{\theta\left( t_{1}^{ccw} \right)} \cdot \cos}\quad{\theta\left( t_{1}^{ccw} \right)}} & 1 \\{2\sin\quad{{\theta\left( t_{2}^{ccw} \right)} \cdot \cos}\quad{\theta\left( t_{2}^{ccw} \right)}} & 1 \\\vdots & \vdots \\{2\sin\quad{{\theta\left( t_{n}^{ccw} \right)} \cdot \cos}\quad{\theta\left( t_{n}^{ccw} \right)}} & 1 \\\ldots & \ldots \\{{- 2}\sin\quad{{\theta\left( t_{1}^{clw} \right)} \cdot \cos}\quad{\theta\left( t_{1}^{clw} \right)}} & 1 \\{{- 2}\sin\quad{{\theta\left( t_{2}^{clw} \right)} \cdot \cos}\quad{\theta\left( t_{2}^{clw} \right)}} & 1 \\\vdots & \vdots \\{{- 2}\sin\quad{{\theta\left( t_{n}^{clw} \right)} \cdot \cos}\quad{\theta\left( t_{n}^{clw} \right)}} & 1\end{bmatrix}\begin{Bmatrix}{R^{2}{\sin(\phi)}} \\R^{2}\end{Bmatrix}}} & (43)\end{matrix}$

Lost motion describes an error that commonly exists in an individualaxis mechanism that is caused by looseness in the mechanical componentsand linkages. When an axis with a lost motion error reverses itsdirection of motion (sign of velocity), the position of the end-pointlags the position of the motor. The A2100 control identifies andcompensates for two classes of lost motion, backlash and windup.

Backlash is a type of lost motion that is present regardless of whetherthe axis is in motion or at rest. That is, it persists after the axishas come to a stop. Windup is a type of lost motion that exists when theaxis is in motion where the axis appears to creep to the correctposition after the motor has stopped. An explanation of the differencesbetween windup and backlash and how they can be compensated is given inU.S. Pat. No. 6,060,854, entitled “Method and Apparatus for Compensatingfor Windup in a Machine,” issued to Stephen J. Yutkowitz, the entiredisclosure of which is incorporated herein by reference.

Both types of lost motion exhibit nearly indistinguishable signatures ona ballbar plot and cannot conventionally be separately identified basedon a single ballbar circle-test measurement. An additional test thatuses the ballbar to allow the two types of lost-motion to bedistinguished from each other has been developed and can be called thebacklash-test. In an exemplary embodiment, the backlash test is aseparate type of test from the circle-test whose measured data has beenthe topic of discussion up to this point. An explanation of how toestimate the total lost motion error from the circle-test data will bepresented first, then an explanation for the backlash test will begiven.

The parametric description of total-lost motion makes the assumptionthat the lost motion parameter does not vary with axis position orvelocity (as long as there are no full stops). Axis lost motion of thebacklash type that does vary with position can be handled (e.g., in theA2100 by the bidirectional compensation capability whose data-tables aredeveloped from laser measurements).

An effect of lost-motion on circular motion is shown in FIG. 12. Thespecific example shown in FIG. 12 is for a case with a very high amountof backlash on the Y-axis and no backlash on the X-axis. Unlike many ofthe previous plots, FIG. 12 is not a polar-error plot; it is a plot ofthe actual circular contour.

The plot of FIG. 12 shows the contour for clockwise motion. The case forcounter-clockwise motion should look like the current plot reflectedabout the Y-axis, as can be imagined. The center-corrected curve issimply the actual curve shifted downward such that its geometric centercorresponds to the center of the ideal circle. The actual y-coordinateof the curve center depends on the direction from which the ballbarcenter fixture was approached when the circle center was defined duringthe fixturing phase of the ballbar test.

Examination of FIG. 12 indicates that the total lost motion errorE_(TLM) causes a flat area following the axis reversal where the motionof the Y-motor does not result in any Y-axis motion while the X-axismoves with no error. This flat area can cause the best-fit circle tohave a radius that is somewhat less than the radius of the programmedcircle. It will be shown that for the type of data generated by anexemplary embodiment of the present invention, the amount of “shrinkage”caused by lost motion should be insignificantly small compared to thetotal lost motion error and need not be considered when determining theabsolute scale errors. The command circle angle that defines thedistance from the reversal where the (Y-axis) motion begins can be wellapproximated by the following equation when the ratio of E_(TLM) to R isfairly small.${\cos(\quad\alpha)} = \frac{R - {\frac{1}{2}E_{TLM\_ Y}}}{R + {\frac{1}{2}E_{TLM\_ Y}}}$

For actual measurements on conventional machine tools, a minimum circleradius can be about 100 mm and the largest expected lost-motion errorshould be well below about 0.2 mm. Thus, the maximum conceivable valuefor the angle α should be less than about 2 degrees. This confirms thatthe amount of shrinkage of the best-fit-circle radius due to lost motionshould be insignificant relative to the lost motion error itself andneed not be considered.

The observation that the flat area, S_(TLM), covers a small percentageof the total circle also suggests that the exact shape of the curve justafter the reversal (henceforth referred to as the transient portion)need not be considered by the technique for estimating the total lostmotion from a ballbar measurement (since, e.g., a built-in requirementcould be that the data arc is relatively large). Nevertheless, a goodapproximate representation of the transient portion can be useful whenattempting to predict how backlash or windup compensation might improvecircularity. The predicted effect of compensation can be obtained bysubtracting a curve developed from the model parameter from themeasurement curve, wherein it is desired to closely match the model tothe actual transient. Thus, for a clockwise circle with a y-axis lostmotion error, the following set of equations applies. $\begin{matrix}{{x(\theta)} = {R\quad{\cos(\theta)}}} & (44) \\{{y_{clw}(\theta)} = \left( \begin{matrix}{{{R\quad{\sin(\theta)}} + {\frac{1}{2}E_{TLM\_ Y}}},} & {{{- \frac{1}{2}}\pi} \leq \theta < {{\frac{1}{2}\pi} - \alpha}} & (a) \\{{R - {\frac{1}{2}E_{TLM\_ Y}}},} & {{{\frac{1}{2}\pi} - \alpha} \leq \theta < {\frac{1}{2}\pi}} & (b) \\{{{R\quad{\sin(\theta)}} - {\frac{1}{2}E_{TLM\_ Y}}},} & {{\frac{1}{2}\pi} \leq \theta < {{\frac{3}{2}\pi} - \alpha}} & (c) \\{{{- R} + {\frac{1}{2}E_{TLM\_ Y}}},} & {{{\frac{3}{2}\pi} - \alpha} \leq \theta < {\frac{3}{2}\pi}} & (d)\end{matrix} \right.} & (45)\end{matrix}$

The equation for the radius squared as a function of circle angle can beobtained by summing the square of the x and y components. The estimationof the total lost motion error may be based on regions (a) and (c) inequation (45) because regions (b) and (d) are very small and should notsignificantly bias the result if their effects are ignored. As apractical matter, the angle α is not known until E_(TLM) is known, sothe transient effects cannot be considered until a first approximationin an iterative method is obtained. The approximate equation for theradius squared can thus be obtained by applying the equation for region(a) to both regions (a) and (b), and by applying the equation for region(c) to both regions (c) and (d). The following equation can be developedby assuming that the lost-motion squared terms are insignificantlysmall. $\begin{matrix}{{r_{clw}^{2}(\theta)} = \left( \begin{matrix}{{R^{2} + {R\quad{\sin(\theta)}E_{TLM\_ Y}}},} & {{{- \frac{1}{2}}\pi} \leq \theta < {\frac{1}{2}\pi}} \\{{R^{2} - {R\quad{\sin(\theta)}E_{TLM\_ Y}}},} & {{\frac{1}{2}\pi} \leq \theta < {\frac{3}{2}\pi}}\end{matrix} \right.} & (46)\end{matrix}$The results for the counter-clockwise circle may be analogously obtainedand can be shown below. $\begin{matrix}{{r_{ccw}^{2}(\theta)} = \left( \begin{matrix}{{R^{2} - {R\quad{\sin(\theta)}E_{TLM\_ Y}}},} & {{{- \frac{1}{2}}\pi} \leq \theta < {\frac{1}{2}\pi}} \\{{R^{2} + {R\quad{\sin(\theta)}E_{TLM\_ Y}}},} & {{\frac{1}{2}\pi} \leq \theta < {\frac{3}{2}\pi}}\end{matrix} \right.} & (47)\end{matrix}$

The equations for estimating the lost motion errors of the X-axis mayalso be obtained using similar techniques or by replacing θ with (θ+½π)in the previous equations. $\begin{matrix}{{r_{clw}^{2}(\theta)} = \left( \begin{matrix}{{R^{2} - {R\quad{\cos(\theta)}E_{TLM\_ X}}},} & {0 \leq \theta < \pi} \\{{R^{2} + {R\quad{\cos(\theta)}E_{TLM\_ X}}},} & {\pi \leq \theta < {2\pi}}\end{matrix} \right.} & (48) \\{{r_{ccw}^{2}(\theta)} = \left( \begin{matrix}{{R^{2} + {R\quad{\cos(\theta)}E_{TLM\_ X}}},} & {0 \leq \theta < \pi} \\{{R^{2} - {R\quad{\cos(\theta)}E_{TLM\_ X}}},} & {\pi \leq \theta < {2\pi}}\end{matrix} \right.} & (49)\end{matrix}$Equations that are analogous to (45) for the counter-clockwise y-axiscase and for both x-axis cases can also be developed, but are not shownhere. The matrix equation for total lost motion presented below includesfactors for lost motion on both axes. In the equation, the notation QI,QII, QIII, QIV are used to represent the circle quadrants (where themeasurement data is segregated into quadrants). $\begin{matrix}{{\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{r^{2}\left( t_{1}^{\quad{ccwQI}} \right)} \\\vdots\end{matrix} \\{r^{2}\left( t_{n_{ccwI}}^{\quad{ccwQI}} \right)}\end{matrix} \\{r^{2}\left( t_{1}^{\quad{ccwQII}} \right)}\end{matrix} \\\vdots\end{matrix} \\{r^{2}\left( t_{n_{ccwII}}^{\quad{ccwQII}} \right)}\end{matrix} \\{r^{2}\left( t_{1}^{\quad{ccwQIII}} \right)}\end{matrix} \\\vdots\end{matrix} \\{r^{2}\left( t_{n_{ccwIII}}^{\quad{ccwQIII}} \right)}\end{matrix} \\{r^{2}\left( t_{1}^{\quad{ccwQIV}} \right)}\end{matrix} \\\vdots\end{matrix} \\{r^{2}\left( t_{n_{ccwIV}}^{\quad{ccwQIV}} \right)}\end{matrix} \\{r^{2}\left( t_{1}^{\quad{clwQI}} \right)}\end{matrix} \\\vdots\end{matrix} \\{r^{2}\left( t_{n_{clwI}}^{\quad{clwQI}} \right)}\end{matrix} \\{r^{2}\left( t_{1}^{\quad{clwQII}} \right)}\end{matrix} \\\vdots\end{matrix} \\{r^{2}\left( t_{n_{clwII}}^{\quad{clwQII}} \right)}\end{matrix} \\{r^{2}\left( t_{1}^{\quad{clwQIII}} \right)}\end{matrix} \\\vdots\end{matrix} \\{r^{2}\left( t_{n_{clwIII}}^{\quad{clwQIII}} \right)}\end{matrix} \\{r^{2}\left( t_{1}^{\quad{clwQIV}} \right)}\end{matrix} \\\vdots\end{matrix} \\{r^{2}\left( t_{n_{clwIV}}^{\quad{clwQIV}} \right)}\end{pmatrix}\begin{bmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{+ {\cos\left( {\theta\left( t_{1}^{\quad{ccwQI}} \right)} \right)}} & {- {\sin\left( {\theta\left( t_{1}^{\quad{ccwQI}} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{+ {\cos\left( {\theta\left( t_{n_{ccwI}}^{\quad{ccwQI}} \right)} \right)}} & {- {\sin\left( {\theta\left( t_{n_{ccwI}}^{\quad{ccwQI}} \right)} \right)}} & 1\end{matrix} \\\begin{matrix}{+ {\cos\left( {\theta\left( t_{1}^{\quad{ccwQII}} \right)} \right)}} & {+ {\sin\left( {\theta\left( t_{1}^{\quad{ccwQII}} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{+ {\cos\left( {\theta\left( t_{n_{ccwII}}^{\quad{ccwQII}} \right)} \right)}} & {+ {\sin\left( {\theta\left( t_{n_{ccwII}}^{\quad{ccwQII}} \right)} \right)}} & 1\end{matrix}\end{matrix} \\\begin{matrix}{- {\cos\left( {\theta\left( t_{1}^{\quad{ccwQIII}} \right)} \right)}} & {+ {\sin\left( {\theta\left( t_{1}^{\quad{ccwQIII}} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{- {\cos\left( {\theta\left( t_{n_{ccwIII}}^{\quad{ccwQIII}} \right)} \right)}} & {+ {\sin\left( {\theta\left( t_{n_{ccwIII}}^{\quad{ccwQIII}} \right)} \right)}} & 1\end{matrix}\end{matrix} \\\begin{matrix}{- {\cos\left( {\theta\left( t_{1}^{\quad{ccwQIV}} \right)} \right)}} & {- {\sin\left( {\theta\left( t_{1}^{\quad{ccwQIV}} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{- {\cos\left( {\theta\left( t_{n_{ccwIV}}^{\quad{ccwQIV}} \right)} \right)}} & {- {\sin\left( {\theta\left( t_{n_{ccwIV}}^{\quad{ccwQIV}} \right)} \right)}} & 1\end{matrix}\end{matrix} \\\begin{matrix}{- {\cos\left( {\theta\left( t_{1}^{\quad{clwQI}} \right)} \right)}} & {+ {\sin\left( {\theta\left( t_{1}^{\quad{clwQI}} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{- {\cos\left( {\theta\left( t_{n_{clwI}}^{\quad{clwQI}} \right)} \right)}} & {+ {\sin\left( {\theta\left( t_{n_{ccwI}}^{\quad{clwQI}} \right)} \right)}} & 1\end{matrix}\end{matrix} \\\begin{matrix}{- {\cos\left( {\theta\left( t_{1}^{\quad{clwQII}} \right)} \right)}} & {- {\sin\left( {\theta\left( t_{1}^{\quad{clwQII}} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{- {\cos\left( {\theta\left( t_{n_{clwII}}^{\quad{clwQII}} \right)} \right)}} & {- {\sin\left( {\theta\left( t_{n_{clwII}}^{\quad{clwQII}} \right)} \right)}} & 1\end{matrix}\end{matrix} \\\begin{matrix}{+ {\cos\left( {\theta\left( t_{1}^{\quad{clwQIII}} \right)} \right)}} & {- {\sin\left( {\theta\left( t_{1}^{\quad{clwQIII}} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{+ {\cos\left( {\theta\left( t_{n_{clwIII}}^{\quad{clwQIII}} \right)} \right)}} & {- {\sin\left( {\theta\left( t_{n_{clwIII}}^{\quad{clwQIII}} \right)} \right)}} & 1\end{matrix}\end{matrix} \\\begin{matrix}{+ {\cos\left( {\theta\left( t_{1}^{\quad{clwQIV}} \right)} \right)}} & {+ {\sin\left( {\theta\left( t_{1}^{\quad{clwQIV}} \right)} \right)}} & 1 \\\vdots & \vdots & \vdots \\{+ {\cos\left( {\theta\left( t_{n_{clwIV}}^{\quad{clwQIV}} \right)} \right)}} & {+ {\sin\left( {\theta\left( t_{n_{clwIV}}^{\quad{clwQIV}} \right)} \right)}} & 1\end{matrix}\end{bmatrix}}\begin{Bmatrix}\begin{matrix}{R \cdot E_{TLM\_ X}} \\{R \cdot E_{TLM\_ Y}}\end{matrix} \\R^{2}\end{Bmatrix}} & (50)\end{matrix}$

An example for a polar-error plot for clockwise and counterclockwisemoves on a machine with lost motion on both X and Y axes is shown inFIG. 13. The post-reversal transient portions for this synthesizedexample are very similar to the post-reversal portions of measurementson actual machines. In the example, the best-fit circle and the idealcircle are so-similar that they are substantially indistinguishable, aswas expected based on the argument of insignificantly small circleshrinkage.

Up to this point, the discussion has been limited to the identificationof the total lost motion from a continuous motion circle-testmeasurement. Experience with these measurements indicates that there isno substantially detectable difference between the effect of thebacklash and windup components of total lost motion. In order toseparately identify each of these errors, an additional backlash testcan be employed.

An exemplary backlash test involves a sequence where the ballbar is usedto detect the lost motion that occurs when the axis moves to positionstops and dwells (as previously described). The windup error for an axiscan then be computed by subtracting the measured backlash from theestimate for total lost motion. If a backlash test is not performed,backlash can be assumed zero and the measured lost motion from thecircle-test attributed to a windup error. For example, this could bebased on the assumption that the machine is properly compensated forbacklash using bidirectional compensation.E _(windup) _(—) _(X) =E _(TLM) _(—) _(X) −E _(backlash) _(—) _(X)

Friction on a machine tool axis can be the source of a transient errorfollowing axis reversals. Currently, for example, the A2100 employsfriction compensation algorithms as part of its integrated feedforwardcapability. The A2100 friction compensation assumes that the frictionforces and torques acting on the axis are reasonably represented by aCoulomb+viscous friction model.

A torque discontinuity due to Coulomb friction occurs after an axisreversal and results in a transient error where the axis lags thecommand for some period of time until the integrator in the velocitycontrol loop builds up to contain a torque that counters the frictiontorque. In general, the size of the transient error increases with thefriction level and decreases with the velocity loop's proportional andintegral gains. A description of how to iteratively tune the frictioncompensation based on a ballbar measurement is described in the A2100Installation Service & Support Manual (referred to hereinafter as the“ISS manual”), which is hereby incorporated by reference.

An effect of Coulomb friction errors on a ballbar plot is an increase inthe circle radius following the reversal for some angle until the servodetects the error and the radius returns to the desired value. Asignature of a friction reversal error on a polar-error plot is shown inFIG. 14.

In contrast to the effect of lost motion, friction errors can cause thebest-fit radius to be slightly larger than the ideal radius. Theamplification of the best fit radius over the ideal radius relative tothe magnitude of the friction reversal error depends mostly on theangular duration of the friction error, which itself is related(primarily) to the velocity loop's integrator time constant and thecircle frequency. To assess the degree to which friction errors may biasthe best fit radius, a worst case scenario could be considered to get anapproximate value for the largest expected angle of the reversal spike.

For example, the anticipated highest circle frequency should occur at aminimum ballbar radius of about 100 mm and a maximum anticipatedfeedrate of about 15,000 mm/min—making the frequency about 2.5 rad/sec.Experience shows that the integrator time constant is typically lessthan about 0.020 sec. The duration of the reversal error is consideredto be significant over two integrator time constants. The angulardistance traveled in two time constants can thus be about 0.040 sec*2.5rad/sec=0.10 rad or 5.7 degrees. Even for the worst case scenario, thefriction error should not significantly amplify the best-fit radiusrelative to the amplitude of the friction error itself.

In another embodiment of the present invention, algorithms forestimating the parameters associated with the friction errors can alsobe included. The results for friction compensation parameters obtainedvia feedforward auto-tuning can often be improved upon by performingsome manual adjustments based on ballbar measurements. The manualadjustments can be typically performed based, for example, on theprocedure outlined in the ISS manual. An explanation for why the resultsof the feedforward auto tuning may be improved by further manual tuningis that the transient friction forces/torques that occur during an axisreversal are more complex than the assumed step discontinuity at zerovelocity. The exact nature of these transient friction torques may bevery complex and may be a function of a number of state variablesincluding the positions, velocities, and accelerations of a number ofbodies in a multi-body system.

This complex behavior may be partially approximated by adjusting theamplitude and time constant of friction compensation (feedforward) todiffer from the ideal values obtained from feedforward auto-tuning'storque measurements and measurements of the response of the velocityloop compensator. This observation has led to the addition of aCoulomb-Width-Adjust parameter to the set of feedforward parameters toenable adjustments of the friction compensation time constant withoutaffecting the other types of feedforward that operate directly on theintegrator time constant (acceleration and viscous frictionfeedforward). Thus, the ballbar measurement may be used to determine anoptimal value for the Coulomb-Width-Adjust parameter, given that thefeedforward-auto-tuning process has already been performed and hasresulted in a value for the estimate of the integrator time constant.

In one embodiment, the ability to estimate friction parameters (e.g.,amplitude and time constant) directly from a measurement of a machinewithout any active friction compensation can be included (e.g., if aprecise estimate of the circle angle for every measurement can beobtained). For example, such an algorithm could first remove knownerror-types from the data using the techniques described below. Next,the radial error in the immediate region of the axis reversal could beconverted to deviation of the reversing axis and the correspondingangles could be converted to the time since reversal. Then, an iterativeprocess can begin by selecting a candidate value for the time constant τ(inverse of integrator gain) and performing a simulation based on thistime constant. A model used to develop the simulation is shown in FIG.15.

The transfer function for the position error in response to a frictionforce signal can be derived from the block diagram in FIG. 15 and can beshown in equation (51) below. The DC amplitude of this transfer functionis arbitrary. The transfer function formulation may be used to develop anumerical solution to the associated linear differential equation.$\begin{matrix}{\frac{e(s)}{f(s)} = \frac{s}{s^{3} + {K_{P}s^{2}} + {{K_{P}\left( {\frac{1}{\tau} + K_{V}} \right)}s} + \frac{K_{V}K_{P}}{\tau}}} & (51)\end{matrix}$

The known axis Kv, an assumption that Je≈J (where Je is an inertiaestimate from feedforward auto tuning and J is actual total axisinertia), and an estimate for the axis Kp (velocity loop proportionalgain) can be used to develop an estimate for the error-transient versustime function of the data sequence for the axis error transientfollowing a reversal. The scaling of such a time sequence might bearbitrary since the friction torque is initially unknown. The value forKp need not be very precise as the pulse shape is quite insensitive tothis parameter (assuming the velocity servo is reasonably well tunedwith a phase margin of more than 40°). The simulation parameter for Kpcan be the same as the iff_kp_estimate.

The amplitude of the estimated transient can be determined using leastsquares to obtain an optimal scaling between the arbitrary-amplitudesimulated transient and the y-axis deviation sequence derived from themeasured radius. The sum of the residual errors squared can be computedand retained. Residual errors can be computed for a number of selectionsfor the time constant as part of an iterative optimization process. Thetime constant and associated amplitude that results in the minimumsum-of-residual-error-squared can be chosen as the estimated frictionparameters. The process can be constructed to simultaneously includeavailable measurements of reversals of the axis under consideration.

A potential problem with the above-described procedure for estimatingthe friction reversal error can be that it might not be valid for caseswhere, for example, the friction compensation is actively applied duringthe ballbar measurement. The ballbar measurement is expected to have themost value in determining an adjustment to nominal values for frictioncompensation parameters obtained via feedforward auto-tuning. Thus, inan exemplary embodiment, a ballbar measurement will be performed undercircumstances where friction compensation is actively applied during themeasurement.

If the friction compensation that is applied is not optimal, then someerror at the reversal might occur, and such an error might be morecomplex than the type produced when friction compensation is turned off.One approach to take in this case could involve attempting to remove theeffects of friction compensation. However, the effects of frictioncompensation are not easily removed from the data because the DC scalingof the transfer function may no longer be arbitrary. This means that a1-dimensional optimization problem could become a 2-dimensionaloptimization problem.

One solution could include automatically applying the iterative manualtuning procedure outlined in the ISS manual. Such an approach coulditeratively execute the following sequence: 1) perform a ballbarmeasurement, 2) isolate the reversal errors, 3) use heuristic rules todetermine whether to increase or decrease the Coulomb-Width-Adjust andwhether to increase or decrease the Coulomb-Friction-Level, 4) perform amodel-based curve fitting to determine candidate values of theseparameters for the next iteration, 5) configure the friction feedforwardwith these new values, and 6) return to step 1. The iterative sequencecan be exited once the errors are below a defined threshold or theimprovements obtained between iterations become insignificant.

Estimation of the friction-related parameters is a capability that couldbe useful for the purpose of data analysis in ranking the impact ofdifferent errors on the overall circularity. It could be used toauto-tune friction compensation for systems wherefeedforward-auto-tuning is unavailable. For systems wherefeedforward-auto-tuning is available, it could, for example, be used topropose modifications to the friction compensation parameters obtainedvia a previously run feedforward-auto-tuning test.

The preceding discussion developed the equations for the individualerror types by assuming that each of these error types was the oneexisting in a measurement. In real applications, the error types occursimultaneously and are present in some proportion. In order to get anoptimal estimate for each error types, the effect of all known errorscan be considered simultaneously. This can be accomplished byformulating a single matrix equation that includes the effects of allknown errors.

The matrix of basis functions can contain, for example, ten columns,with each column containing a basis function for one of the followingexemplary parameters:

-   -   1. X axis center offset.    -   2. Y axis center offset    -   3. XY Squareness Error    -   4. XY Scale-mismatch error    -   5. XY Servo mismatch    -   6. X Straightness Error    -   7. Y Straightness Error    -   8. X Total Lost Motion    -   9. Y Total Lost Motion    -   10. Best Fit Radius

The matrix equation can be formulated by simply combining each of thematrix equations previously discussed. Since the servo-mismatch andlost-motion estimate includes separate treatment of data based on circledirection, the data can be segregated into clockwise and counterclockwise segments. Additionally, the total lost motion estimateinvolves separate treatment of data based on quadrant, with the databeing further partitioned based on circle quadrant (estimate angle).Partitioning of this type has already been described during thediscussion of lost motion estimation as seen in equation (50).

A combined matrix equation for simultaneous estimation of all parametersis shown below. To enable the equation to fit on the page, the datapartitioning is not explicitly shown (the ± symbols are used todesignate matrix columns whose basis function depends on directionand/or quadrant). $\begin{matrix}{\left\{ {r^{2}\left( t_{i} \right)} \right\} = {\begin{bmatrix}\left\{ {{- 2}{\cos\left( {\theta\left( t_{i} \right)} \right)}} \right\}^{T} \\\left\{ {{- 2}{\sin\left( {\theta\left( t_{i} \right)} \right)}} \right\}^{T} \\\left\{ {{\cos\left( {\theta\left( t_{i} \right)} \right)}{\sin\left( {\theta\left( t_{i} \right)} \right)}} \right\}^{T} \\\left\{ {2\left( {{\cos^{2}\left( {\theta\left( t_{i} \right)} \right)} - {\sin^{2}\left( {\theta\left( t_{i} \right)} \right)}} \right)} \right\}^{T} \\\left\{ {{\pm 2}{\cos\left( {\theta\left( t_{i} \right)} \right)}{\sin\left( {\theta\left( t_{i} \right)} \right)}} \right\}^{T} \\\left\{ {2{\sin\left( {\theta\left( t_{i} \right)} \right)}{\cos^{2}\left( {\theta\left( t_{i} \right)} \right)}} \right\}^{T} \\\left\{ {2{\cos\left( {\theta\left( t_{i} \right)} \right)}{\sin^{2}\left( {\theta\left( t_{i} \right)} \right)}} \right\}^{T} \\\left\{ {\pm {\cos\left( {\theta\left( t_{i} \right)} \right)}} \right\}^{T} \\\left\{ {\pm {\sin\left( {\theta\left( t_{i} \right)} \right)}} \right\}^{T} \\\left\{ {1\quad 1\quad\ldots\quad 1} \right\}^{T}\end{bmatrix}^{T} \cdot \left( \begin{matrix}{R \cdot E_{CTR\_ X}} \\{R \cdot E_{CTR\_ Y}} \\{R^{2} \cdot E_{SQR\_ XY}} \\{R^{2} \cdot E_{SCL\_ XY}} \\{R^{2} \cdot {\sin\left( \phi_{svo} \right)}} \\{R^{3} \cdot E_{{STR\_ X}{\_ Y}}} \\{R^{3} \cdot E_{{STR\_ Y}{\_ X}}} \\{R \cdot E_{TLM\_ X}} \\{R \cdot E_{TLM\_ Y}} \\R^{2}\end{matrix} \right\}}} & (52)\end{matrix}$The following symbolic matrix equation can be used to describe theprevious specific equation.r ²(t)_((n×1)) =[A] _((n×10)) ·{P} _((10×1))  (53)

The estimation of the above ten parameters can be accomplished byapplying the least squares solution of equation (3) to equation (52)above. In the following equations the ^ symbol is used to designate aparameter as being an estimated value.{circumflex over (P)}=[A] ^(T) [A] ⁻¹ [A] ^(T) r ²  (54)

The inversion of the matrix A^(T)A may be accomplished using eigenvaluesand eigenvectors obtained by Jacobi's technique. Using the Jacobitechnique has the advantage that it provides the eigenvalues of[A^(T)A], which are used in the computation of the condition number asdescribed below.

Once the matrix is inverted, the scaling of the parameter by powers ofthe best fit radius can be removed by multiplying each parameter by theappropriate power of the inverse of the best-fit radius obtained bysquare-rooting the final element of the parameter vector.$\begin{matrix}\begin{matrix}{{\hat{R} = \sqrt{{\hat{P}}_{10}}},} & {{E_{CTR\_ X} = \frac{{\hat{P}}_{1}}{\hat{R}}},} & {{E_{CTR\_ Y} = \frac{{\hat{P}}_{2}}{\hat{R}}},} & \quad \\{{E_{SQR\_ XY} = \frac{{\hat{P}}_{3}}{{\hat{R}}^{2}}},} & {{E_{SCL\_ XY} = \frac{{\hat{P}}_{4}}{{\hat{R}}^{2}}},} & {{\phi_{svo} = {\sin^{- 1}\left( \frac{{\hat{P}}_{5}}{{\hat{R}}^{2}} \right)}},} & \quad \\{{E_{STR\_ XY} = \frac{{\hat{P}}_{6}}{{\hat{R}}^{3}}},} & {{E_{STR\_ YX} = \frac{{\hat{P}}_{7}}{{\hat{R}}^{3}}},} & {{E_{TLM\_ X} = \frac{{\hat{P}}_{8}}{\hat{R}}},} & \quad \\{{E_{TLM\_ Y} = \frac{{\hat{P}}_{9}}{\hat{R}}},} & \quad & \quad & \quad\end{matrix} & (55)\end{matrix}$

Examination of the matrix [A] in equation (52) reveals that there areopportunities in the formulation for some columns to be redundant withother columns. If a column is not unique, then the parameter associatedwith that column cannot be uniquely identified and the least squaressolution to the matrix equation fails to exist. The columns can be madeunique by ensuring that the data arc is sufficiently large and that bothclockwise and counter-clockwise data is present.

An indicator of the overall uniqueness of the columns in the matrixformulation is the matrix condition number (the ratio of the maximumsingular value to the minimum singular value). The singular values maybe obtained by computing the square root of the eigenvalues of the[A]^(T)[A]] matrix (from equation (3)). The condition number for a givenset of test conditions can be normalized based on the (best case)condition number for a full 360 degree data arc in both directions.

A statistic called the “excitation” is defined as the inverse of thenormalized condition number. A test for the sufficiency of excitationprovided by a collection of test setup parameters might be to compareexcitation statistic to a threshold. When an inappropriate set of testconditions is proposed by the user, the excitation falls below apredefined threshold and the test conditions can be rejected.$\begin{matrix}{{\left\{ \lambda \right\} = {{eig}\left\lbrack {\lbrack A\rbrack^{T}\lbrack A\rbrack} \right\rbrack}},{C_{NUM} = \sqrt{\frac{\lambda_{\max}}{\lambda_{\min}}}},{{excitation} = \frac{C_{NUM}}{C_{NUM}\left( {{{360^{\circ}\quad{clw}}\&}{ccw}} \right)}}} & (56)\end{matrix}$

A plot of the excitation as a function of data arc length for a data arcstarting at 0.0 is shown in FIG. 16. Similar plots for other (non-zero)data start angles are very similar to the one shown in FIG. 16. The plotindicates that the best excitation is obtained for a data arc whoselength is greater than about 280 degrees. The minimum acceptableexcitation allowed for by the test setup validity test can be chosen tobe about 0.2 or approximately 220 degrees of data arc.

Some data analysis tasks provided as described by example herein involvecomputing a synthesized value for the ballbar radius for each angle atwhich a ballbar radius measurement has been taken. The sequence ofsynthesized radii can be referred to as r_(synth)(t). The synthesizedradius sequence can be used by the following analyses: 1) modelingcoherence, 2) ranking the relative impact of each error type on theoverall circularity error, 3) predicting how the circularity error wouldchange if a selected combination of axis compensations were employed,and/or 4) estimating the measurement variance used to determine thesignificance of each estimated parameter.

The synthesized radius vector can be obtained by evaluating equation(52), where the parameter vector is populated with the estimates, andperforming the square-root of the result. The synthetic data can then bemodified at the axis reversals to account for the lost motion transientsof the type described by equation (45)b and (45)c. The data followingthe reversal can be modified by computing the radius for a constant axisdeviation over the region defined by the angle α. The followingequations describe how the synthetic data can be generated.$\begin{matrix}{\left\{ {r_{synth}(\theta)} \right\} = \sqrt{\lbrack A\rbrack \cdot \left\{ \hat{P} \right\}}} & (57) \\{{{\hat{\alpha}}_{X} = {\cos^{- 1}\left( \frac{\hat{R} - {\frac{1}{2}{\hat{E}}_{TLM\_ X}}}{\hat{R} + {\frac{1}{2}{\hat{E}}_{TLM\_ X}}} \right)}},{{\hat{\alpha}}_{Y} = {\cos^{- 1}\left( \frac{\hat{R} - {\frac{1}{2}{\hat{E}}_{TLM\_ Y}}}{\hat{R} + {\frac{1}{2}{\hat{E}}_{TLM\_ Y}}} \right)}},} & (58) \\{{r_{synth}(\theta)} = \left\{ \begin{matrix}\sqrt{{{\hat{R}}^{2}{\sin^{2}(\theta)}} + \left( {\hat{R} - {\frac{1}{2}{\hat{E}}_{{TLM}_{X}}}} \right)^{2}} & {\forall{{{{clw}\text{:}}\quad - {\hat{\alpha}}_{X}} \leq \theta < {{0\quad{or}\quad\pi} - {\hat{\alpha}}_{X}} \leq \theta < \pi}} & \quad \\\sqrt{{{\hat{R}}^{2}{\sin^{2}(\theta)}} + \left( {\hat{R} - {\frac{1}{2}{\hat{E}}_{{TLM}_{X}}}} \right)^{2}} & {\forall{{{ccw}\text{:}\quad 0} \leq \theta < {{\hat{\alpha}}_{X}\quad{or}\quad\pi} \leq \theta < {\pi + {\hat{\alpha}}_{X}}}} & \quad\end{matrix} \right.} & (59) \\{{r_{synth}(\theta)} = \left\{ \begin{matrix}\sqrt{{{\hat{R}}^{2}{\cos^{2}(\theta)}} + \left( {\hat{R} - {\frac{1}{2}{\hat{E}}_{{TLM}_{Y}}}} \right)^{2}} & {\forall{{{{clw}\text{:}\quad\frac{\pi}{2}} - {\hat{\alpha}}_{Y}} \leq \theta < {{\frac{\pi}{2}\quad{or}\quad\frac{3\pi}{2}} - {\hat{\alpha}}_{Y}} \leq \theta < \frac{3\pi}{2}}} & \quad \\\sqrt{{{\hat{R}}^{2}{\cos^{2}(\theta)}} + \left( {\hat{R} - {\frac{1}{2}{\hat{E}}_{{TLM}_{Y}}}} \right)^{2}} & {\forall{{{ccw}\text{:}\quad\frac{\pi}{2}} \leq \theta < {\frac{\pi}{2} + {{\hat{\alpha}}_{Y}\quad{or}\quad\frac{3\pi}{2}}} \leq \theta < {\frac{3\pi}{2} + {\hat{\alpha}}_{Y}}}} & \quad\end{matrix} \right.} & (60)\end{matrix}$In the previous equations, the value for mean radius was for the casewhere all errors, except for the lost motion under consideration, arepresent in the data set.

The sequence of synthesized radii can be used to demonstrate what isexpected to occur in the absence of certain error types or combinationsof error types. As described earlier, for example, analyze-data page 70can allow a user to select a combination of error types to be removedfrom the measurement (e.g., by checking or unchecking checkboxes todesignate the desired combination). For example, when a checkbox ischecked, the associated error type can be included in thepredicted-radius polar-error plot. When it is unchecked, the effect ofthe corresponding error can be removed. In this way, the user can get arealistic idea of how application of compensation can improveperformance by removing the known error effects from measured data thatmay contain error types for which no compensation is available.

The first step in constructing the predicted radius sequence can be tocompute the residual error by subtracting elements of the synthesizedradius sequence from corresponding elements in the measured radiussequence. The residual error can then be added to a second synthesizedradius sequence r_(synth) _(—) _(partial) that can be created by zeroingall elements of the {P} vector except for those whose error types remain(e.g., those checked on diagnostics window 102). If an axis lost motionerror is included, then modification of the synthesized sequence can beperformed by employing equations (59) and (60) for the axis or axeswhose lost motion error should be included in the prediction sequence.

In an exemplary embodiment, checkboxes for the center offset errors arenot provided because the center offsets are not machine errors and thepredicted radius sequence should have the effect of center offsetsremoved. This means that, for the computation of r_(synth) _(—)_(partial) the first two elements of {P} can be set to zero. A procedurefor computing the predicted radius r_(predicted) is now explicitlydescribed.

First, the residual error between the synthesized radius and themeasurement can be computed.

 e _(r) =r−r _(synth)  (61)

Next, the synthesized radius including only the error types whosecheckboxes are checked can be computed.

In the following equation, each element of the vector {c} can beassigned a value of 0.0 when the checkbox associated with the parameteris checked, and a value of 1.0 when the checkbox associated with theparameter is not checked. $\begin{matrix}{\left\{ {r_{synth\_ part}\left( {\theta\left\{ c \right\}} \right)} \right\} = \sqrt{\left\lbrack {A(\theta)} \right\rbrack \cdot \left\{ \left| 0 \middle| 0 \middle| {c_{3}{\hat{P}}_{3}} \middle| {c_{4}{\hat{P}}_{4}} \middle| {c_{5}{\hat{P}}_{5}} \middle| {c_{6}{\hat{P}}_{6}} \middle| {c_{7}{\hat{P}}_{7}} \middle| {c_{8}{\hat{P}}_{8}} \middle| {c_{9}{\hat{P}}_{9}} \middle| {\hat{P}}_{10} \right| \right\}^{T}}} & (62)\end{matrix}$

If c₇=1, use equation (59) to apply the x-axis reversal transientmodification to r_(synth) _(—) _(part). Likewise, if c₈=1, use equation(60) to apply the y-axis reversal transient modification to r_(synth)_(—) _(part). Finally, the predicted radius for the selected combinationof error types can be computed.r _(predicted)(θ,{c})=r _(synth) _(—) _(part)(θ,{c})+e _(r)(θ)  (63)

The predicted radius sequence from the previous equation has no centeroffset error. It can be used directly in the computation of thecorrected circularity displayed, for example, in the diagnostics window102.

A capability that can be provided by diagnostics window 102 can includedisplaying and ranking error types in order of their relativecontribution to the circularity error. Each error type's contributioncan be determined by evaluating equation (62) for r_(synth) _(—)_(part)(θ,{c}) where the elements in the vector {c} are set to zero,except the one associated with the particular error type underconsideration (set to 1.0). The circularity error for the resultingsynthesized radius can be computed and retained.

The process can be performed for each of the, in this embodiment, sevenerror parameters. The circularity errors associated with each error typecan be sorted and displayed when the user requests. The process can besummarized by the following equation, which makes use of the Kronickerdelta δ(k): a 10 element vector whose elements are zero, except for thek'th element which has a value of one.E _(CIRC)(k)=maxr _(synth) _(—) _(part) θ,{c}=δ(k)−minr _(synth) _(—)_(part) θ, {c}=δ(k), k=3, 4, . . . , 9  (64)

A statistic called the “model coherence” can be used to indicate whetherthe data is well approximated by the model. It can be used to identifysituations where error sources that are beyond the scope of the assumedmodel may be present, or situations where the measurement data wascorrupted for some reason. A coherence parameter can be obtained bycomputing the square of the coefficient of correlation between themeasured radius vector to the synthesized radius vector r_(synth) thatcontains the effects of all known errors.

In an exemplary embodiment, the center offset errors are removed fromboth radius vectors prior to computation of coherence. Additionally, thecontiguous data segments that follow an axis reversal over the angularranges of α_(x) and α_(y) can be removed from both the measured andsynthesized radius. One advantage associated with eliminating these dataregions can include avoiding the inclusion of much or all of thefriction reversal error transient, which can lower the coherence, in thecomputation.

As the coherence deviates from the ideal value of 1.0, the ability ofthe model to represent the observed behavior becomes potentiallyquestionable. Typically, a coherence value above about 0.90 indicates agood fit. Accordingly, in an exemplary embodiment, a warning is given(e.g., displayed) if the coherence is unacceptably low to avoidsituations where a less experienced user might attempt to compensate amachine based on a bad measurement.

One problem with a definition of coherence based on correlationcoefficient can be that, when the machine has little or no errors, orwhen the errors are well compensated, the presence of random noise oraxis vibration may cause the computed coherence to be low. If the samemachine had some errors present that are within the scope of the model,the coherence would have been high. One solution to such a problem couldinvolve performing analyses to distinguish between components of theresidual error that appear to be random relative to significant changesin circle angle to those residual errors that appear to have adeterministic low-frequency relationship to circle angle. Evaluation ofthe autocorrelation function of the residual error or applying anangle-domain smoothing filter to the residual error could potentiallyidentify the existence of non-random signal content.

Another solution involves intentionally adding some random noise to boththe measured and the synthesized data in order to provide somecorrelation for the case where there are no errors present. Theamplitude of this intentional random noise should be low enough to allowfor true problems with the measurement to be identified with a lowcoherence, but high enough to prevent low coherence statistics frombeing reported for error-free data. Thus, the amplitude should be basedon the expected residual error caused by vibration and measurementnoise.

Coherence can be computed based on the following equations where ρ² isthe coherence, η(t) is a sequence of uniformly distributed random noisewith a predefined range, the “˜” overstrike is used to designate asignal that has been intentionally augmented with the addition of noise,the “-” overstrike is used to designate the mean value of a sequence,and the “synth” subscript is used to identify the synthesized datasequence. Experience has shown that a good value for the range of theuniformly distributed random variable η is about ±10 microns.{tilde over (r)} _(synth)(t)=r _(synth)(t)+η(t)  (65){tilde over (r)}(t)=r(t)+η(t)  (66)

Next, these data sequences can be modified by computing the centeroffset error and removing its effects from the data by applyingequations (6) and (7) to both data sets. Otherwise, a large centeroffset error may result in a good coherence even if there aremeasurement problems. Finally, the correlation coefficient can becomputed for the relationship for the center-corrected, noise-augmentedsequences for measured and synthesized radius. $\begin{matrix}{\rho^{2} = \frac{\left( {\sum{\left( {{{\overset{\sim}{r}}_{synth}(t)} - \overset{\_}{{\overset{\sim}{r}}_{synth}}} \right) \cdot \left( {{\overset{\sim}{r}(t)} - \overset{\_}{\overset{\sim}{r}}} \right)}} \right)^{2}}{\sum{\left( {{{\overset{\sim}{r}}_{synth}(t)} - \overset{\_}{{\overset{\sim}{r}}_{synth}}} \right)^{2} \cdot {\sum\left( {{\overset{\sim}{r}(t)} - \overset{\_}{\overset{\sim}{r}}} \right)^{2}}}}} & (67)\end{matrix}$

“Significance” is a statistic that can be computed for each estimatedparameter. It is designed to indicate, for example, whether themagnitude of the parameter estimate is significantly large relative theestimated uncertainty of the parameter. For example, the parameterestimate for lost-motion on a machine with zero backlash is likely to benon-zero due to the presence of measurement noise. In this case, thecomputed significance for the total-lost-motion parameter should besmall and the user can appreciate that the estimate for the total lostmotion is not particularly meaningful.

The equations for the computation of significance can be developed basedon several assumptions about the data. These assumptions are notnecessarily totally valid, yet the significance statistic developedbased on these assumptions has proven to be useful. The significancestatistic compares the parameter estimate to an estimate of thestandard-deviation associated with the parameter. The estimate of thestandard deviation of the parameter can be computed based on theassumption that the error model is capable of perfectly describing themeasurement, with the exception of normally distributed, uncorrelatedrandom noise of constant variance on the radius-squared measurement.

Implicit within the stated assumption is that the basis functions thatconstitute the matrix [A] in equation (54) are deterministic (i.e., theestimates for the circle angle based on the elapsed test time contain noerrors). With these assumptions, the covariance matrix for the parameterestimate vector {{circumflex over (P)}} in equation (54) may bedetermined from the equation shown below, whose derivation can be foundin many textbooks whose topic is statistics and estimation-theory.

The following equation computes the exemplary 10×10 covariance matrix,where N is the total number of measurements and 10 is the number ofparameters estimated. As with the estimation of coherence, the dataimmediately following the reversals is not included in the computationin order to assure that the transient reversal errors do not bias theestimate of the measurement noise variance. $\begin{matrix}{\left\lbrack {COV}_{P} \right\rbrack = {\left( {\frac{1}{N - 10}{\sum\limits_{i = 1}^{N}\left( {r^{2} - {\lbrack A\rbrack\left\{ \hat{P} \right\}}} \right)^{2}}} \right) \cdot \left\lbrack {\lbrack A\rbrack^{T}\lbrack A\rbrack} \right\rbrack^{- 1}}} & (68)\end{matrix}$

The estimated standard deviation of the estimated parameters is thesquare-roots of the diagonal elements of the covariance matrix.{{circumflex over (σ)}_(P)}=√{square root over ({diag([COV_(P)])})}  (69)Each component of the standard deviation vector can be compared to thevalue of the corresponding component in the parameter estimate vector{{circumflex over (P)}} to determine whether the estimated value issignificant relative to the standard deviation. The formulation for thesignificance statistic can be based on the following statement: thesignificance increases as the ratio of parameter-estimate to standarddeviation increases, and a parameter estimate whose amplitude is lessthan, for example, five standard deviations is considered insignificant.An equation for the significance S of the k'th element of the parameterestimate vector developed to meet the stated criteria is shown below.$\begin{matrix}{S_{k} = \left\{ \begin{matrix}{{1 - \frac{\left( {5 \cdot {\hat{\sigma}}_{P\_ k}} \right)}{{{\hat{P}}_{k}}}},} & {{\hat{P}}_{k} \geq {5 \cdot {\hat{\sigma}}_{P\_ k}}} \\{0,} & {{\hat{P}}_{k} < {5 \cdot {\hat{\sigma}}_{P\_ k}}}\end{matrix} \right.} & (70)\end{matrix}$

The previous equations provide a general approach for computing thesignificance. Some special consideration for the scale error estimatesfor the case of a length-calibrated ballbar can be included as theseparameters are not explicitly contained in the parameter estimatesvector and can be developed based on two separate parameters, each withits own variance estimate. When a simplifying assumption that theaverage radius is much greater than the variance of the estimate for theaverage radius is employed, the following expression may be obtainedfrom equation (25).var({circumflex over (R)} ² ·Ê _(SCL) _(—) _(X))=var({circumflex over(R)} ² ·Ê _(SCL) _(—) _(Y))≈({circumflex over (σ)}_(P) _(—) ₄)²  (71)

Equations for the significance for the axis scale errors are shownbelow. $\begin{matrix}{{S\left( {\hat{E}}_{SCL\_ X} \right)} = \left\{ \begin{matrix}{{1 - \frac{\left( {5 \cdot {\hat{\sigma}}_{{P\_}4}} \right)}{\left( {{\hat{R}}^{2} \cdot {\hat{E}}_{SCL\_ X}} \right)}},} & {{{\hat{R}}^{2} \cdot {\hat{E}}_{SCL\_ X}} \geq {5 \cdot {\hat{\sigma}}_{{P\_}4}}} \\{0,} & {{{\hat{R}}^{2} \cdot {\hat{E}}_{SCL\_ X}} < {5 \cdot {\hat{\sigma}}_{{P\_}4}}}\end{matrix} \right.} & (72) \\{{S\left( {\hat{E}}_{SCL\_ Y} \right)} = \left\{ \begin{matrix}{{1 - \frac{\left( {5 \cdot {\hat{\sigma}}_{{P\_}4}} \right)}{\left( {{\hat{R}}^{2} \cdot {\hat{E}}_{SCL\_ Y}} \right)}},} & {{{\hat{R}}^{2} \cdot {\hat{E}}_{SCL\_ Y}} \geq {5 \cdot {\hat{\sigma}}_{{P\_}4}}} \\{0,} & {{{\hat{R}}^{2} \cdot {\hat{E}}_{SCL\_ Y}} < {5 \cdot {\hat{\sigma}}_{{P\_}4}}}\end{matrix} \right.} & (73)\end{matrix}$

Although the exemplary development of the significance statistic isbased on an assumption that model residual errors were uncorrelated,normally-distributed random-variables, it has proven to be a usefulstatistic even for cases where the residual error is caused by somedeterministic phenomenon that is not included within the scope of themodel structure. The computed significance for each error parameter canbe used by the update compensation algorithms to determine whether ornot to update a proposed value for a particular compensation parameterrelative to the existing compensation. If the significance for aparameter is zero, then the proposed value for compensation can bechosen to be the existing original value. Otherwise, it can be developedbased on the parameter estimate as described below.

Additionally, the significance of the parameter estimate associated witheach proposed compensation value can be graphically displayed (e.g., asmeter 150 on adjust compensation page 112). This can, for example,provide the user with some additional information to aid in the decisionof whether to use a proposed value for compensation.

The model structure for machine tool errors employed as specificallydisclosed by example herein addresses a limited range of possibilities.Many other error types are known to exist that are not currently handledby this particular example (but which one of ordinary skill in the artcould appreciate could be handled using the teachings of the presentinvention). Some of the unmodeled errors might be static, while othersmight be dynamic.

With respect to analysis of ballbar measurements, some unmodeled errortypes may be falsely attributed to one or more of the errors that arecontained within the model structure, while other error types do not fitwithin the model structure and their presence results in a reduced valueof the coherence statistic. Some unmodeled errors are parametric innature (i.e., they can be represented by a limited number of parameters)while others are not suitable for parameterization. Some known machineerror types that are not addressed by the structure of the machine errormodel of the exemplary embodiments as specifically disclosed by exampleherein are listed below.

False Attribu- tion in Suitable For Exemplary Para- Dynamic/ Modelmeteriza- Error Description Static Structure? tion? Feedback orMechanism Static No Yes Kinematics Cyclic Errors Torque Ripple CyclicErrors Dyn No Potentially Complex Linear Axis Straightness Static NoProbably not Errors Simple Axis Angular Errors (Roll, Static PotentiallyYes Pitch, Yaw) Complex Axis Angular Errors Static No Probably not

Of the error types listed above, the simple axis angular errors may, forexample, be falsely attributed as linear squareness, straightness orscale errors under the exemplary embodiments discussed herein. Simpleaxis angular errors refer to errors in the orientation of the machinemember that vary linearly or parabolically as a function of axisposition.

For example, there can be a total of nine angular error types (e.g.,roll, pitch, and yaw errors for each of three linear direction axes).Each error type can be associated with an associated error amplitudeparameter and two parameters associated with the location of therotation axis (e.g., resulting in a total of 27 parameters). If bothlinear and parabolic varieties of these error types are present, thenthere can be a total of 54 possible angular error parameters. Errors ofthis type not only potentially effect the relative orientation of, forexample, a tool tip to a workpiece, but can also create displacementerrors between the tool tip and workpiece.

The displacement errors created by angular errors may appear verysimilar to the linear axis squareness and straightness errors containedwithin the assumed exemplary error model structure. This can be seen byexamining FIG. 17, showing the displacement errors created by just oneof the nine error types.

Referring to FIG. 17, it can be seen that a Y-axis roll error creates adisplacement error with components in the X and Z directions. Thedisplacement error depends on the location in the machine enveloperelative to the roll axis. For a ballbar measurement, the effect of theangular errors should appear to be very similar to YX squareness, YZsquareness, YX straightness, and YZ straightness errors.

Attribution of angular errors to the linear squareness, straightness,and scale error parameters is not necessarily a detrimental effect.Parameters for squareness straightness and scale error compensationdeveloped based on displacements that originally was caused by angularerrors will likely improve the accuracy of the machine as long as themachine motion is in or near the plane of the ballbar test. Any attemptto incorporate these “simple” angular errors into the assumed exemplarymodel structure might involve estimating error parameters bysimultaneously analyzing measurement results obtained from multipleballbar measurements in multiple machine orientation planes and instacked parallel planes.

The A2100 control, for example, has the ability to compensate for mostof the error types previously identified herein. The compensationalgorithms can be parameterized in much the same way as the errorsthemselves, with some minor differences that will be discussed below.Typically, the compensation parameter has an inverse relationship withthe error type.

For example, if the straightness error coefficient is positive, then thecompensation parameter is negative. The error parameters can beestimated for the Cartesian horizontal and vertical axes within thecircle plane, but the compensation is applied to a physical axis or axesthat actuate motion of the logical axis that corresponds to thehorizontal or vertical axis. If the test is performed with some amountof compensation active, then the estimated error parameter defines theincremental adjustment to the existing parameter.

Accordingly, an exemplary embodiment of the present invention hasknowledge of the configuration of each compensation type for eachphysical axis involved in the motion within the circle plane. Thus,conversion of an error parameter estimate to a compensation parameterinvolves mapping the existing compensation from data associated to aphysical axis (or axes) to the circle plane-axis (horizontal orvertical), computing the new value of compensation based on theparameter estimate for the circle-plane axis, then mapping the newcompensation value to the physical axis or axes.

The discussions regarding the analysis of measured data referred to thex and y axes. The equations can be interpreted in such a way that xdesignates the horizontal axis and y designates the vertical axis. Inthis way, the equations that have been developed for x and y can beapplicable to the axes in any of the machine's major planes.

The major planes can be identified by naming the horizontal axisfollowed by the vertical axis (e.g., XY, XZ, YZ). The equationsdiscussed herein require no modification for the XY plane, and may beapplied to, for example, the YZ plane by substituting x for y and y forz.

Unlike the other two planes, the horizontal and vertical axes of the XZplane describe a left-handed coordinate system and, for squareness andstraightness coefficients, a sign change accompanies the substitution.This plane is defined by the left-handed XZ coordinates rather than theright-handed ZX coordinates due to a previously established conventionfor machine tools. The discussions that follow use designators h and vfor horizontal and vertical axes within the circle-plane, and p todesignate the third axis perpendicular to the plane.

It has already been seen that configuration data for the physical axisthat produces motion of a circle-plane axis is involved when computingan error parameter for the circle plane axis. Such a mapping fromphysical axis to circle-plane axis can be anticipated by equations (28)through (30) that account for the expected servo effects on the averagecircle radius. These equations can be used when computing the overallaxis scale errors for a test involving a length-calibrated ballbar.

The aforementioned equations need knowledge of the circle plane axesvalues for Kv_(nominal), Kv_(adjust), and velocity feedforward (VFF).These parameters can be data items associated with the physical axes.The composite Kv (Kv_(nominal),* Kv_(adjust)) and the applied VFF forthe circle plane axis can be stored in, for example, the test resultdata-file's header (the applied VFF can be the configured VFF percentagefor the axis if test-setup-data's feed-mode has anapply_rate_feedforward attribute that is true and is zero otherwise). Asystem level assumption can be that the machine's linear axes havematched gains and VFF levels.

Ideally, values for Kv and VFF can be obtained by examiningconfiguration data from one physical axis that actuates one of thecircle-plane axes. In practice, the Kv and VFF are included for each ofthe circle-plane axes in the data file header. Thus, a data file headeraccording to an exemplary embodiment of the present invention contains aKv_(h) a Kv_(v) a VFF_(h) and a VFF_(v).

The evaluation of equations (28) through (30) uses each parameter'saverage value for the plane. This is considered reasonable because asignificant mismatch between axes can be identified by parameterestimation, and a user would be expected to correct the situation andre-perform the test. Another complicating factor can be the possibilitythat a logical axis may be mapped to one or more physical axes. Thissituation occurs for split and slave axes.

In order to accommodate for both split and slave axes, the configurationdata item for the logical axis can be obtained from the data for thefirst actively normal physical connection at the time of the test. Thisis reasonable for split axes because the configuration sets for each ofthe physical connections are expected to match each other and it isanticipated that compensation be applied equally to the physical axesthat make up the split set. For slave axes, it is anticipated that eachaxis may be compensated separately. This can be accomplished byperforming a separate ballbar test for each slave axis while the otherslave axes are removed. Since the other slave axes have a status ofremoved (i.e. not normal) during the test, the data set for the firstnormal connection can be the appropriate choice.

Similar considerations as those discussed for Kv and VFF can be madewhen obtaining the existing compensation for an axis. As mentioned, thedevelopment of proposed modification to compensation based on a ballbartest measurement can include: 1) the estimate of the parametric error;and 2) the values for compensation parameters that were active in themachine at the time the test was performed. As with the retention of theKv and VFF, in an exemplary embodiment, the compensation present on themachine during the test may be stored in the data file header. In analternative embodiment, rather than store the compensation in the datafile header, a unique text string, the data reference number, can bewritten to the file header.

The data reference number can be developed based on the time and date atthe beginning of a ballbar test run. If the data reference number storedin memory for the most recent test matches the data reference numberfrom the file header for the measurement data currently loaded intomemory, then the proposed values for compensation for the circle planeof the loaded measurement can be computed based on the existingcompensation parameters from the data base. Once changes to thecompensation are applied, the data reference number in memory can becleared and the display of proposed values can be terminated (thecompensation values present during the test are no longer accuratelyrepresented by the current values in the database).

The existing parameters for scale, squareness, and straightnesscompensation from the data for the first physical connection to thelogical-axis associated with the circle-plane axis can be employed asthe original compensation values. This approach can be taken toaccommodate both split and slave axes as described above. Once thecompensation is computed, it can be applied uniformly to normalconnections to the logical axis. This can be done if the exemplaryembodiment described herein is the primary method for developing thesecompensation parameters.

Uniform application to all axes making up the split set can be areasonable choice given that measurements of positions of individualphysical axes are not performed. The same approach can be taken forbacklash and windup compensation; but, in this case it is conceivablethat the exemplary embodiment might detrimentally over-write values forbacklash and windup that are intentionally set to different values foreach axis in the split set based on a more specialized measurement. Inan exemplary embodiment as described herein, however, backlash andwindup compensation are typically applied uniformly to all axes in asplit set based on the results of a ballbar or similar measurement oflogical axis end point.

Associations between axes for straightness/squareness compensation canbe based on physical axes and limited to four compensation axes perindependent physical axis. The straightness/squareness compensationfacility can also be used to compensate for scale errors by compensatingan axis relative to itself. For a simple configuration where each linearaxis (e.g., X, Y, Z) is actuated by a single physical axis, threeassociations for each physical axis can be sufficient to compensate eachcross axis for straightness and squareness errors in the direction ofthe cross axis and to compensate the axis itself for scale mismatcherrors.

For the case where there is one split axis actuated by two physicalaxes, four associations per independent axis are utilized. For example,consider a case where the x axis is a split with two physical axes, andthe y and z axes are actuated by a single physical axes. Compensationrelative to the x-axis position can be accomplished by associating the yand z axes to the first x connection for squareness and straightness, athird association for the first connection can be consumed incompensating for scale error, and the second connection uses just oneassociation (to itself) to compensate for scale error. Compensationrelative to the y (or z) axis consumes all available associations (e.g.,each of the two physical connections to x consume one association, zconsumes another, and the self-association for scale compensationconsumes the fourth).

As is implied by the previous example, associations forsquareness/straightness can be allocated separately from associationsfor scale-error. The algorithm for assigning associations attempts tocreate the cross-axis associations by using the first connection tological as the independent axis. A cross axis association can be usedwhen either squareness compensation or straightness compensation for theindependent axis is enabled.

When an attempt is made to apply the proposed values of compensation,existing associations for each physical axis can be first scannedthrough to identify any existing associations of the type required andthe parameters associated with these associations updated. If there isno existing association of the type required, a new one can be created.If all available associations for an axis are consumed, an alert can bereported instructing the user to delete any extra associations, such asthose that involve a rotary axis.

Proposed values for compensation parameters are typically computed bysubtracting the computed error parameter from the value for the originalcompensation parameter (i.e. the value of the parameter that was used toactively apply compensation during execution of the circle-test).Proposed values can be computed for the axes involved in the plane ofthe most recent circle test. The following equations summarize howcompensation can be developed from an estimated error parameter for thehorizontal or vertical axis. It should be understood that the equationsare applied to each physical connection to the logical axes associatedwith the h or v axis for the particular circle plane.

Squareness/straightness compensation can be defined by the followingfour parameters: 1) the squareness comp coefficient, 2) the straightnesscomp coefficient, 3) the compensation origin, and 4) the compensationrange. A procedure for developing the compensation parameters from themeasurement results is outlined below. In the equations that describethe compensation parameters, the axes to which squareness orstraightness compensation applies are identified by the subscripts wherethe first subscript identifies the independent axis and the secondsubscript identifies the dependent axis. For example, sqr_comp_(HV) isthe coefficient of squareness compensation that adjusts the position ofthe V axis as a function of the position of the H axis.

Equation (52) provides the parameter E_(SQR) _(—) _(HV) that describesthe squareness error of the two axes that make up the circle plane.However, squareness compensation parameters can be available for bothaxes. Compensation can be applied to the vertical axis as a function ofthe position of the horizontal axis and compensation to the horizontalaxis can be applied to the horizontal axis as a function of the verticalaxis position.

For the sake of symmetry, both available compensations can be employedby equally distributing the squareness error to each availableassociation. For the case of the XZ plane, the sign of the squarenesserror can be negated to account for the fact that the equation wasdeveloped for a right-handed coordinate system, but H=X, V=Z is a lefthanded coordinate system. The equations can be generalized for any ofthe three coordinate systems by defining a sign-adjust variable that isa function of the circle plane. $\begin{matrix}{{sgn\_ adj} = \left\{ \begin{matrix}{{+ 1},} & {{XY}\quad{Plane}} \\{{- 1},} & {{XZ}\quad{Plane}} \\{{+ 1},} & {{YZ}\quad{Plane}}\end{matrix} \right.} & (74)\end{matrix}$

A general equation for squareness compensation adjustment is shownbelow. $\begin{matrix}\begin{matrix}{{sqr\_ comp}_{HV}^{proposed} = {{sqr\_ comp}_{HV}^{original} - {\frac{1}{2}{{sgn\_ adj} \cdot E_{SQR\_ HV}}}}} \\{{sqr\_ comp}_{VH}^{proposed} = {{sqr\_ comp}_{VH}^{original} - {\frac{1}{2}{{sgn\_ adj} \cdot E_{SQR\_ HV}}}}}\end{matrix} & (75)\end{matrix}$When considering squareness compensation, the compensation origin can bearbitrary since it defines the location of the independent axis fromwhich to measure displacement. However, this same parameter can bere-used by straightness compensation and, in that case, the straightnesserror can be computed as if the compensation origin is equal to theindependent axis coordinate of the circle center.

The proposed value for straightness compensation can be developeddirectly from the original value for the coefficient and the estimatedstraightness error if the independent axis coordinate of the circlecenter for the test from which the error parameters are estimated is thesame (or nearly the same) as the compensation origin from the originaldata. The following equation, for example, may be employed. In theequation below, the sign of the estimated straightness coefficient ismodified for the X axis in the XZ coordinate system. $\begin{matrix}\begin{matrix}{{str\_ comp}_{HV}^{proposed} = {{str\_ comp}_{HV}^{original} - {{sgn\_ adj} \cdot E_{STR\_ HV}}}} \\{{str\_ comp}_{VH}^{proposed} = {{str\_ comp}_{VH}^{original} - E_{STR\_ HV}}}\end{matrix} & (76)\end{matrix}$

Straightness compensation limits the amount of compensation applied tobe the compensation applied at a distance from the compensation originthat is equal to the compensation range. Accordingly, the compensationorigin can be matched to the circle center for the recent test or, forthe case of partial circles, be set to the coordinate that representsthe midpoint of the axis range of motion experienced during the test.The proposed value of the compensation range can be the radius of thetest circle. This “clipping” can help avoid large errors that can beintroduced when compensation is extrapolated to regions outside of therange of the measurement. The clipping behavior is illustrated by FIG.18.

If the clipping did not occur, then changes to the circle center couldbe accommodated by making adjustments to both squareness andstraightness compensation. Given the presence of the clipping, theeffect of straightness compensation present in the measured data can beremoved from the data prior to performing the estimate for theparametric straightness error. The resulting estimate could then be useddirectly (with a possible sign adjustment) as the straightnesscompensation coefficient. Such a consideration can also be used forcases when the original compensation origin matches the circle center,but the radius of the circle test is greater than the originalcompensation range.

Another solution to the changing center problem involves having testsetup validation algorithms test whether the selected values for circlecenter coordinates are near the compensation origin for the currentlyconfigured straightness compensation. If the circle center issignificantly different from the compensation origin, or if the circleradius is significantly larger than the compensation-range, then thetest setup validation can issue a message informing the user that, ifthe test results are to be used to develop squareness/straightnesscompensation, then the straightness compensation for the axis or axeswhose circle center does not match the compensation origin should bedisabled prior to performing the test.

A threshold to identify a significant difference between circle centercoordinate and original compensation origin coordinate can be about 20%of the circle radius. Likewise, a threshold for a significant differencebetween the circle radius and the original compensation range can bewhen the circle radius is about 20% greater than the originalcompensation range. With these precautions in place, equation (76) maybe applied directly.

Since squareness compensation varies based on the arbitrary choice ofcircle 110 center, both types of compensation might be clipped ornot-clipped together. The decision of whether or not to clip acompensation might be configurable for each compensation type for eachassociation.

Scale error compensation can be implemented Via, for example, thestraightness/squareness compensation capability of a control (e.g., theA2100) by associating the axis to itself and populating the squarenesscompensation parameter with a parameter derived from the estimated valueof the scale error. The straightness coefficient for theself-association can be zeroed and the enable Boolean for straightnesscan be set to off. Thus, the proposed value for scale compensation canbe developed using the following equation. $\begin{matrix}\begin{matrix}{{{sqr\_ comp}_{HH}^{proposed} = {{sqr\_ comp}_{HH}^{original} - E_{SCL\_ H}}},{{str\_ comp}_{HH}^{proposed} = 0}} \\{{{sqr\_ comp}_{VV}^{proposed} = {{sqr\_ comp}_{VV}^{original} - E_{SCL\_ V}}},{{str\_ comp}_{VV}^{proposed} = 0}}\end{matrix} & (77)\end{matrix}$

If the ballbar is not length calibrated, then the compensation can beadjusted to account for scale mismatch (equation (24) would have beenused to compute the estimate for scale error). This might be reasonableif the original value for scale compensation was developed based on ameasurement with a length-calibrated ballbar. However, if the scaleerror has changed significantly since the development of the originalcompensation, or if the scale error compensation is originally developedwith the ballbar length not-calibrated, then the scale errorcompensation can probably only adjust for mismatches between axes withinthe plane.

As tests (with the ballbar not length-calibrated) are performed formultiple planes, the scale mismatch compensation that worked for oneplane will be detrimentally over-written by the scale mismatchcompensation for another plane. In such a situation, the order in whichthe per-machine-plane tests are arranged can affect the value for anaxis' scale compensation, and a compensation coefficient that corrects amismatch in one plane can conceivably aggravate a mismatch in the otherplane. This problem should not occur when all tests are performed with alength calibrated ballbar, as the result for an axis's proposed valuefor scale error compensation should be the same (correct value) fortests performed in either of the two machine major-planes in which theaxis lies. The associated exemplary embodiment helps the user avoid thissituation by issuing a warning message when the user attempts to copythe proposed values for scale error compensation to the active valuescolumn if the proposed value was developed based on measurements wherethe ballbar was not length calibrated.

Straightness compensation could be used in a self-association to correctfor screw-pitch errors that are not constant everywhere on the screw,but where the pitch error increases in an approximate linearrelationship to screw position. If such cases are found on actualmachine tools, then augmentation of the existing capability couldinclude a parameterized variable pitch error using the straightnesscoefficient.

Backlash compensation and windup compensation are both represented aspositive numbers for positive windup and backlash errors.$\begin{matrix}\begin{matrix}{{backlash\_ comp}_{H}^{proposed} = {{backlash\_ comp}_{H}^{original} + E_{backlash\_ H}}} \\{{backlash\_ comp}_{V}^{proposed} = {{backlash\_ comp}_{V}^{original} + E_{backlash\_ V}}}\end{matrix} & (78) \\\begin{matrix}{{windup\_ comp}_{H}^{proposed} = {{windup\_ comp}_{H}^{original} + E_{windup\_ H}}} \\{{windup\_ comp}_{V}^{proposed} = {{windup\_ comp}_{V}^{original} + E_{windup\_ V}}}\end{matrix} & (79)\end{matrix}$Both types of lost motion are errors that occur on the axis. They arenot expected to vary with circle plane and can be compensated per-axis.It is expected that the estimated value for axis lost motion should bethe same regardless of which of the two available circle-planes was usedto perform the measurement.

One capability that can be provided by an exemplary embodiment of thepresent invention could involve ensuring that a user constructs a set oftest conditions that are within the capability of the machine andcapable of providing usable results. For example, validity checks can beperformed on user-entered test-setup parameters and combinations ofparameters. These validity checks were previously summarized in theprevious discussion regarding test setup page 10.

Exemplary equations employed for these checks are herein provided. Atest for the validity of measured data that compares the total test timeto the expected test time is also described here. The data validity testcan be very similar to the overshoot arc length test and a design for asingle utility (e.g., software) employed by both tests is presented.

In an exemplary embodiment, a circle center is within the range limitsof the machine axes. The test setup displacement coordinates can bespecified using the machine coordinates for the axis. The test-setupconfiguration values for circle center can be directly compared to themachine configuration data for axis range limits.

If the following equations are true, the validity test should pass.X_(min) designates the lower limit for the x-axis range (LOW_LIMT[X]),X_(max) designates the high limit for the x-axis range (HIGH_LIMT[X]),etc.X _(min) ≦X _(CIRC) _(—) _(CTR) ≦X _(max)Y _(min) ≦Y _(CIRC) _(—) _(CTR) ≦Y _(max)Z _(min) ≦Z _(CIRC) _(—) _(CTR) ≦Z _(max)  (80)

As previously described with respect to the discussion regarding testsetup page 10, the circle radius can be less than or equal to theballbar length, but not so small that motion in the direction of theaxis perpendicular to the circle plane contributes significantly to theballbar measurement. In an exemplary embodiment, a change in ballbarlength dL, relative to a change in position in the direction of theperpendicular axis dP can be chosen that is less than half the change inballbar length relative to a change in position along a radial with thecircle plane dR. $\begin{matrix}\left. \left( {\frac{\mathbb{d}L}{\mathbb{d}P} \leq {\frac{1}{2}\frac{\mathbb{d}L}{\mathbb{d}R}}} \right)\Rightarrow\left( {P \leq {\frac{1}{2}R}} \right) \right. & (81)\end{matrix}$The previous equation specifies a maximum cone angle φ_(max).$\begin{matrix}{{\tan\left( \phi_{\max} \right)} = {\left. \frac{P - P_{CTR}}{R} \right|_{\max} = {\left. \frac{1}{2}\Rightarrow\phi_{\max} \right. = {{\tan^{- 1}\left( \frac{1}{2} \right)} = 22.6^{\circ}}}}} & (82)\end{matrix}$

A circle radius relative to the ballbar length based on the maximum coneangle is shown below.L cos(φ_(max)=22.6°)≦R0.9·L≦R  (83)Finally a requirement for circle radius can be:0.9·L _(BALLBAR) ≦R _(CIRCLE) ≦L _(BALLBAR)  (84)

Since this check involves knowledge of the ballbar radius, in anexemplary embodiment, it is not performed until the user attempts toapply a proposed set of test configuration parameters. In an exemplaryembodiment, the ballbar length is not subject to a validity test becauseits configured value is limited to a finite number of selections in adrop-down menu uicontrol (described further herein). The values in thedrop down menu can be nominal lengths of the ballbar, plus the availableextension bars and combinations of extension bars.

A previous discussion regarding equations for simultaneous estimation ofthe error parameters describes how the parameter estimation algorithmdistinguishes one error type from another based on the excitation of themeasurement. The excitation increases with the length of the data arc,but can also depend on the angle of the data arc start point. A test ofwhether the data arc is long enough to provide sufficient excitation foruniquely identifying the various error parameters can be to compute theexcitation parameter using equation (56). In an exemplary embodiment,the excitation is required to exceed the value for a 180° data arcstarting at an angle of 45° relative to the horizontal axis. Theexcitation can be computed by constructing an [A]-matrix from equations52 and 53 with clockwise and counter-clockwise data covering the samedata-arc range. The condition for a data arc and start angle providingsufficient excitation is represented by equation (85) below. The angularspacing used to construct the [A]-matrix in equation (85) can be thegreater of 1° or the predicted angle change based on federate and radiusas represented by equation (86) below.Excitcition([A](θ_(data) _(—) _(start),θ_(data) _(—)_(range),Δθ))≧Excitation([A](45°,180°,1°))  (85)

-   -   where: $\begin{matrix}        {{\Delta\quad\theta} = {\max\left( {{\frac{1}{f_{sample}} \cdot \frac{180}{\pi} \cdot \frac{Feedrate}{60 \cdot R_{CIRCLE}}},1.0^{\circ}} \right)}} & (86)        \end{matrix}$

In an exemplary embodiment, tests for ensuring that the circle test willnot cause an axis range limit to be exceeded simultaneously consider thecircle center, circle radius, ballbar length, start angle ofacceleration arc, and total arc length. An exemplary test is designed toensure that the total circle from the beginning of the accelerationovershoot arc to the end of the deceleration overshoot arc nowhereinvolves exceeding a range limit of an axis. If the test discovers thata range limit might be exceeded, changes to the circle center and/orcircle radius bringing the test to within range can be proposed.

A constant height of the perpendicular P axis can be checked. The testsetup parameter for the P coordinate of the circle center is not reallythe circle center—it is the P coordinate of the center ball. In anexemplary embodiment, the perpendicular coordinate of the true circlecenter is located at or “above” the entered value for circle center intest setup data. Thus, in the following equation, the square root isadded to the configured circle center (not subtracted). $\begin{matrix}{P_{\min} \leq \left( {P_{CTRC\_ CTR} + \sqrt{L_{ballbar}^{2} - R_{circle}^{2}}} \right) \leq P_{\max}} & (87)\end{matrix}$

Once the perpendicular axis position is found to be within range, themaximum and minimum axis locations during the circular motion can bedetermined. These values can be evaluated by first determining thebounds for a circle whose total arc(acc-overshoot-arc+data-arc+dec-overshoot-arc) covers 360°, and thenrefining these bounds by considering the specific case if the total arcis less than 360°. The bounds can be initially computed for circlecenter coordinates of (0,0), and the actual coordinates can beconsidered after having made any refinements for a partial circle.H _(max) =R _(circle) , H _(min) =−R _(circle) , V _(max) =R _(circle) ,V _(min) =−R _(circle)  (88)

If the total circle (overshoot-arcs+data-arc) spans an arc of 360°, thenthere need not be any additional refinement of axis bounds. Otherwise,the next step can be to identify the quadrant in which the arc beginsand the quadrant in which the arc ends, identify any quadrant boundariesthat are crossed, and re-evaluate the bounds that correspond to quadrantboundaries that are not crossed. This process might not easily berepresented by a single equation since it involves a large degree oflogic and unique handling for each of the 16 possible combinations ofstart-quadrant and end quadrant according to an exemplary embodiment.The start and end quadrants can be determined by establishing the unitvector from circle center to acc-arc-start point and the unit vectorfrom circle center to dec-arc-stop point. $\begin{matrix}{{{\hat{u}}_{start} = \begin{Bmatrix}{\cos\left( {\theta_{data\_ start} - \theta_{overshoot}} \right)} \\{\sin\left( {\theta_{data\_ start} - \theta_{overshoot}} \right)}\end{Bmatrix}},{{\hat{u}}_{end} = \begin{Bmatrix}{\cos\left( {\theta_{data\_ start} + \theta_{data\_ length} + \theta_{overshoot}} \right)} \\{\sin\left( {\theta_{data\_ start} + \theta_{data\_ length} + \theta_{overshoot}} \right)}\end{Bmatrix}}} & (89)\end{matrix}$

The start and end quadrants can be determined from the unit vectors asfollows: $\begin{matrix}{Q_{start} = \left\{ {\begin{matrix}{I,} & {{u_{start}^{H} > 0},} & {u_{start}^{V} \geq 0} \\{{II},} & {{u_{start}^{H} \leq 0},} & {u_{start}^{V} > 0} \\{{III},} & {{u_{start}^{H} < 0},} & {u_{start}^{V} \leq 0} \\{{IV},} & {{u_{start}^{H} \geq 0},} & {u_{start}^{V} < 0}\end{matrix},{Q_{end} = \left\{ \begin{matrix}{I,} & {{u_{end}^{H} > 0},} & {u_{end}^{V} \geq 0} \\{{II},} & {{u_{end}^{H} \leq 0},} & {u_{end}^{V} > 0} \\{{III},} & {{u_{end}^{H} < 0},} & {u_{end}^{V} \leq 0} \\{{IV},} & {{u_{end}^{H} \geq 0},} & {u_{end}^{V} < 0}\end{matrix} \right.}} \right.} & (90)\end{matrix}$The maximum and minimum axis displacements can be refined based on theparticular combination of start quadrant and end quadrant for each ofthe 16 combinations shown in FIG. 19. Exemplary equations for eachcombination are provided below. $\begin{matrix}\left\{ \begin{matrix}{\begin{matrix}\begin{matrix}Q_{start} \\ = \end{matrix} \\I\end{matrix}\left\{ \begin{matrix}{\begin{matrix}{Q_{end} = {I\&}} \\\left( {u_{start}^{H} > u_{end}^{H}} \right)\end{matrix}\text{:}} & {H_{\max} = {R \cdot u_{start}^{H}}} & {H_{\min} = {R \cdot u_{end}^{H}}} & {V_{\max} = {R \cdot u_{end}^{V}}} & {V_{\min} = {R \cdot u_{start}^{V}}} \\{Q_{end} = {{II}\text{:}}} & {H_{\max} = {R \cdot u_{start}^{H}}} & {H_{\min} = {R \cdot u_{end}^{H}}} & {V_{\max} = R} & {V_{\min} = {R \cdot \left\{ {u_{start}^{V},u_{end}^{V}} \right\}_{MIN}}} \\{Q_{end} = {{III}\text{:}}} & {H_{\max} = {R \cdot u_{start}^{H}}} & {H_{\min} = {- R}} & {V_{\max} = R} & {V_{\min} = {R \cdot u_{end}^{V}}} \\{Q_{end} = {{IV}\text{:}}} & {H_{\max} = {R \cdot \left\{ {u_{start}^{H},u_{end}^{H}} \right\}_{MAX}}} & {H_{\min} = {- R}} & {V_{\max} = R} & {V_{\max} = {- R}}\end{matrix} \right.} \\{\begin{matrix}\begin{matrix}Q_{start} \\ = \end{matrix} \\{II}\end{matrix}\left\{ \begin{matrix}{Q_{end} = {I\text{:}}} & {H_{\max} = R} & {H_{\min} = {- R}} & {V_{\max} = {R \cdot \left\{ {u_{start}^{V},u_{end}^{V}} \right\}_{MAX}}} & \quad \\{\begin{matrix}{Q_{end} = {{II}\&}} \\\left( {u_{start}^{V} > u_{end}^{V}} \right)\end{matrix}\text{:}} & {H_{\max} = {R \cdot u_{start}^{H}}} & {H_{\min} = {R \cdot u_{end}^{H}}} & {V_{\max} = {R \cdot u_{start}^{V}}} & {V_{\min} = {R \cdot \cdot u_{end}^{V}}} \\{Q_{end} = {{III}\text{:}}} & {H_{\max} = {R \cdot \left\{ {u_{start}^{H},u_{end}^{H}} \right\}_{MAX}}} & {H_{\min} = {- R}} & {V_{\max} = {R \cdot u_{start}^{V}}} & {V_{\min} = {R \cdot u_{end}^{V}}} \\{Q_{end} = {{IV}\text{:}}} & {H_{\max} = {R \cdot u_{end}^{H}}} & {H_{\min} = {- R}} & {V_{\max} = {R \cdot u_{start}^{V}}} & {V_{\min} = {- R}}\end{matrix} \right.} \\{\begin{matrix}\begin{matrix}Q_{start} \\ = \end{matrix} \\{III}\end{matrix}\left\{ \begin{matrix}{Q_{end} = {I\text{:}}} & {H_{\max} = R} & {H_{\min} = {R \cdot u_{start}^{H}}} & {V_{\max} = {R \cdot u_{end}^{V}}} & {V_{\min} = {- R}} \\{Q_{end} = {{II}\text{:}}} & {H_{\max} = R} & {H_{\min} = {R \cdot \left\{ {u_{start}^{H},u_{end}^{H}} \right\}_{MIN}}} & {V_{\max} = R} & {V_{\min} = {- R}} \\{\begin{matrix}{Q_{end} = {{III}\&}} \\\left( {u_{start}^{V} > u_{end}^{V}} \right)\end{matrix}\text{:}} & {H_{\max} = {R \cdot u_{end}^{H}}} & {H_{\min} = {R \cdot u_{start}^{H}}} & {V_{\max} = {R \cdot u_{start}^{V}}} & {V_{\min} = {R \cdot u_{end}^{V}}} \\{Q_{end} = {{IV}\text{:}}} & {H_{\max} = {R \cdot u_{end}^{H}}} & {H_{\min} = {R \cdot u_{start}^{H}}} & {V_{\max} = {R \cdot \left\{ {u_{start}^{V},u_{end}^{V}} \right\}_{MAX}}} & {V_{\min} = {- R}}\end{matrix} \right.} \\{\begin{matrix}\begin{matrix}Q_{start} \\ = \end{matrix} \\{IV}\end{matrix}\left\{ \begin{matrix}{Q_{end} = {I\text{:}}} & {H_{\max} = R} & {H_{\min} = {R \cdot \left\{ {u_{start}^{H},u_{end}^{H}} \right\}_{MIN}}} & {V_{\max} = {R \cdot u_{end}^{V}}} & {V_{\min} = {R \cdot u_{start}^{V}}} \\{Q_{end} = {{II}\text{:}}} & {H_{\max} = R} & {H_{\min} = {R \cdot u_{end}^{H}}} & {V_{\max} = R} & {V_{\min} = {R \cdot u_{start}^{V}}} \\{Q_{end} = {{III}\text{:}}} & {H_{\max} = R} & {H_{\min} = {- R}} & {V_{\max} = R} & {V_{\min} = {R \cdot \left\{ {u_{start}^{V},u_{end}^{V}} \right\}_{MIN}}} \\{\begin{matrix}{Q_{end} = {{IV}\&}} \\\left( {u_{start}^{H} < u_{end}^{H}} \right)\end{matrix}\text{:}} & {H_{\max} = {R \cdot u_{end}^{H}}} & {H_{\min} = {R \cdot u_{start}^{H}}} & {V_{\max} = {R \cdot u_{end}^{V}}} & {V_{\min} = {R \cdot u_{start}^{V}}}\end{matrix} \right.}\end{matrix} \right. & (91)\end{matrix}$

After evaluation of the equations from the appropriate row above, thecircle axis ranges can be tested against the available machine axisrange to determine whether the machine envelope is large enough toaccommodate the specified circle. If the machine is not large enough,and a change to the center coordinates can bring the circle within themachine envelope, a proposed change to the center coordinate can beprovided. The process can be represented by the following pseudo-codefor the case of the H axis—an analogous procedure can be implemented forthe V axis.

IF: (H_(CTR) + H_(MAX)) > HIGH_LIMIT[H] OR (H_(CTR) + H_(MIN)) <LOW_LIMIT[H] MESSAGE: “defined circle parameters will cause H axis toexceed a range limit..” IF: (HIGH_LIMIT[H] - LOW_LIMIT[H]) < (H_(MAX)−H_(MIN)) MESSAGE: “chosen circle radius is too large for H axis range”ELSE: H_(CTR1) = LOW_LIMIT[H] + ½(H_(MAX) −H_(MIN)) H_(CTR2) =LOW_LIMIT[H] − ½(H_(MAX) −H_(MIN)) MESSAGE: “valid H axis range forcircle center: {H_(CTR1)}<H<{H_(CTR2)}” ENDIF ENDIFThe messages seen by the operator contain the appropriate axis letterassociated with the H or V axis rather than the H or V as shown above.The terms inside the braces “{ }” above will contain numeric values.

Assuming that it has been established that the circle is within themachine range, then one final test can be to ensure that the positionprior to an initial (e.g., 1.5 mm) feed-in and the position following afinal (e.g., 1.5 mm) feed-out are also within range. These positions canbe defined by the following equations. $\begin{matrix}{\phi = {\tan^{- 1}\left( \frac{P - P_{CTR}}{R} \right)}} & (92)\end{matrix}$  P _(feedin) =P _(feedout) =P _(CTR)+(L_(ballbar)+1.5)·sin(φ)  (93)H _(feedin) =H _(CTR)+(L _(ballbar)+1.5)·cos(φ)·cos(θ_(data) _(—)_(start)−θ_(overshoot))  (94)V _(feedin) =V _(CTR)+(L _(ballbar)+1.5)·cos(φ)·sin(θ_(data) _(—)_(start)−θ_(overshoot))  (95)H _(feedout) =H _(CTR)+(L _(ballbar)+1.5)·cos(φ)·cos(θ_(data) _(—)_(start)+θ_(data) _(—) _(length)+θ_(overshot))  (96) V _(feedout) =V _(CTR)+(L _(ballbar)+1.5)·cos(φ)·sin(θ_(data) _(—)_(start)+θ_(data) _(—) _(length)+θ_(overshoot))  (97)If one of these coordinates is outside of the associated axis' rangelimit, then the validity check can issue a warning to inform the user ofthe situation.

A check for sufficient overshoot arc length can be used to ensurethat: 1) the acceleration overshoot arc is large enough to allow thepath speed to attain the programmed feedrate during the accelerationfrom the speed at the end of the initial feed in; and 2) thedeceleration overshoot arc is long enough such that deceleration to thespeed prior to the final feed-out move begins some time during thedeceleration arc—not during the data arc. The test for the overshoot arclength can consider the programmed feedrate, the circle radius, the dataarc start and end angles, the control's velocity algorithms, and themachine configuration parameters for the axes and path rates. Anexemplary embodiment of the present invention can also include theability to predict a distance consumed to change speed from the feedrateat the end of the feed-in to the programmed feedrate and a distance tochange speed from the programmed feedrate to the feedrate just beforethe final feed-out move.

In one embodiment, such checks can be performed through off-lineemulation of the part program (or parts thereof). For example, theexecution of the entire part program (e.g., feed-in, accelerationovershoot arc, data arc, deceleration overshoot arc, and feed-put) couldbe emulated and analyzed to determine if the aforementioned conditionsare met. In one such embodiment, for example, a velocity versuscumulative distance traveled profile generated by the emulation could beanalyzed to ensure such conditions are met (velocity versus time andcumulative distance traveled versus time profiles could also be used).For example, the velocity at the cumulative distances that represent thebeginning and ending of the data arc could be analyzed to determine ifthey are at the full programmed feedrate. Moreover, such profiles (e.g.,a cumulative distance traveled versus time profile) could also beanalyzed to estimate the total test time (e.g., by tagging data pointscorresponding to expected cumulative distances for the feed-in andfeed-out triggers) and data arc time (e.g., tagging data pointscorresponding to cumulative distances representing the beginning and endof the data arc).

Alternatively, as most of the calculations in such an emulation will beduring the data arc (a full feedrate portion), and are thereforeunnecessary, such checks could be performed by emulating the executionof selected portions of the part program. For example, execution of thepart program could be emulated from just prior to the feed-in move untilthe target feedrate is achieved. The corresponding cumulative distancetraveled could be analyzed to determine if it is less than thecumulative distance that represents the beginning of the data arc(indicating that the acceleration overshoot arc should be sufficient).Moreover, a data point corresponding to such a distance could be taggedand the emulated execution continued until the end of the accelerationovershoot arc is reached, wherein that data point is also tagged and canbe used to determine the total time to perform the accelerationovershoot arc (which could be used with respect to extraction of thedata arc).

Through symmetry, the time for the deceleration overshoot arc can beassumed to be reasonably similar to the time for the accelerationovershoot arc. A separate check can, however, also be performed on thesufficiency of and time to perform the deceleration overshoot arc. Forexample, emulation (with the initial condition of full feedrate) can bestarted at some distance before the start of the deceleration arc (e.g.,at a distance early enough before the start of the deceleration arc toreasonably assume that motion is at the full feedrate). Theaforementioned assumption based on symmetry can be used to determinesuch a distance. The feedrate can be monitored during the emulation anda data point tagged when the feed rate first reduces. A data pointcorresponding to the cumulative distance associated with the beginningof the deceleration overshoot arc can also be tagged. If the feedratestarted to reduce after the beginning of the deceleration overshoot arc,then the deceleration overshoot arc is sufficient.

Further according to one such embodiment, the emulation can continuerunning until the end of the program (the feed-out move). Accordingly,the amount of time of the deceleration overshoot arc can also bedetermined (e.g., from the data point indicating the beginning of thedeceleration overshoot arc and a data point corresponding to feed-outmove). Total test time can also be determined by, for example, addingthe time for the acceleration overshoot arc, the time for thedeceleration overshoot arc, and the time for the data arc (which can bedetermined, for example, by dividing the data arc length by thefeedrate). Although separate software could be developed to perform theaforementioned emulation, in one embodiment of the present invention, autility program can be created that calls the real time velocity managersoftware from the non-real time process to perform the emulation.

Data from a ballbar measurement can be stored, for example, in a diskfile. Facilities for loading historic files and performing analysis onthese historic files can be provided. Files can be of a format thatcould be defined by a ballbar manufacturer, or by the CNC manufacturer.This enables analysis of measurements taken using software from theballbar manufacturer.

Referring by example to the Renishaw QC10 ballbar, it is a device thatplugs into a serial port of a computer. An abstract interface object isused to access and control the ballbar device. This software interfaceprovides methods for initializing the corn port, powering on theballbar, setting the ballbar's measurement coefficients, and startingdata collection. The data collection is performed in an execution threadthat is a member of the process that initializes the ballbar softwareobject. The ballbar deflection is sampled at a constant frequency (e.g.,250 Hertz). The data is copied from the serial port to a data pipe. Thedata is stored as floating point values that represents the signeddeflection of the ballbar nominal deflection in units of mm. The pipecan retain some maximum number of measurements. The ballbar deviceobject generates an event whenever the pipe contains new data. Theballbar device has publicly visible attributes that describe the statusof the ballbar and reports any error conditions. An event is generatedwhen the ballbar object detects an error condition.

The methods and techniques employed by an associated exemplaryembodiment of the present invention to realize the execution of theballbar measurement test as previously described with respect to, forexample, run test page 30 are now further described. The steps forperforming the physical setup and the performance of the actualmeasurement test can be handled by, for example, a Run-Test-Coordinatorfinite state machine (RTC FSM). One of the states of the RTC FSM can bethe TestInProgress state, which itself can involve execution of a nestedFSM called the TestInProgress (TIP) FSM. In an exemplary embodiment, theFSMs and the program execution on the CNC real time process interact ina timely manner, allowing for a smooth and responsive interface as seenby the user (not clunky and jerky).

Safeguards to ensure proper sequencing of events can be provided toaccount for the possibility of highly variable processor loads. Thesegoals can be achieved by polling, for example, the CNC real-time data,invoking the RTC FSM, and letting the FSM run until it reaches its exitpoint. The process can be accomplished by recurrently invoking the FSMactivities method from an autonomous thread within a non-real timeenvironment. A parallel thread buffers the measurement data by movingdata from the ballbar pipe into a dynamically allocated data bufferwhose size is based on the estimated maximum test duration. In this way,the ballbar pipe can be kept as empty as possible in order to avoid pipeoverflow and data loss. If a pipe overflow condition is detected, thenthe FSM transitions to an error state and generates the appropriatealert.

The RTC FSM may be invoked by, for example, an external requirementoriginating from the human operator or by a recurrent invocationgenerated as a background task associated with an exemplary embodimentof the present invention. An invocation originating from a humanoperator may be a user interface (UI) control callback event caused by,for example, pushing a button on the screen, an acknowledgement of analert dialog, or by a reset event that occurs when the user changes thetest-plane, changes the test-setup data, or enters a Run-Test screen forthe first time. Any invocation of any FSM method requires that theclient obtain a mutex to the FSM.

The execution of the state machines can be driven by, for example, datafrom the real time process, events from the ballbar device, and userinputs via the user interface controls (Uicontrols) from a screen. Ifthese data sources are developed from within multiple threads orprocesses, the RTC FSM can be designed to run within a multi-threadedenvironment. Threads/processes involved in an exemplary test realizationare described by FIG. 20, which indicates that five threads/processescan be involved in test realization, including:

-   -   1. Ballbar Device Thread—an autonomous thread running under the        OS (e.g., Windows) that involves collecting data from the        ballbar via the serial port, processing the data, and copying        the measurement to, for example, a pipe object. For example, the        ballbar device software object provided by Renishaw furnishes        the interface for establishing this thread and reporting the        ballbar status.    -   2. Empty Pipe Thread—a high priority thread running under the OS        that copies data from the pipe into a data buffer of dynamically        allocated memory whose length can be based on the anticipated        test duration. This thread empties the pipe when it receives an        event signal from the ballbar device that new data: is present        in the pipe. The data can be stored in, for example, a buffer        object: “resamp_pipe_buffer.” This data can be accessed by the        RTC FSM when evaluating state transition stimulus conditions.        This thread may also invoke the RTC FSM when certain triggering        conditions are detected. These triggering conditions are the        ballbar entering the measurement range from the fully extended        position, entering the fully extended position from the        measurement range, and entering the fully contracted position.        The full measurement data-set can be accessed when performing        data analysis (e.g., curve fitting).    -   3. FSM Ping Thread—the recurrent invocation of the state        activities method of the RTC FSM. The invocations originate from        the user interface display. The interval between attempted        recurrent invocation of the FSM can be small to keep the test        moving along and the priority of this thread can be lower than        that of the other threads discussed.    -   4. Uicontrol Thread—the thread that responds to the operator's        inputs via on-screen user interface controls. Certain Uicontrol        callback functions cause an invocation of the RTC FSM's        respond-to-stimulus method. Access to the RTC FSM methods can be        mutexed such that a request for services from the Ping Thread        waits until the Uicontrol thread's request is completed and        vice-versa. In an exemplary embodiment uicontrol requests are        not queued; a new request replaces any existing request that is        pending.    -   5. CNC Program Execution Process—the program execution process        can be the real time autonomous process that runs under the        CNC's real time OS. The run test FSM makes decisions about when        and whether to perform a state transition based on data from the        real time process. A semaphore (referred to as “waitinloop”)        communicates with a mechanism that allows the RTC FSM to        exercise a limited degree of control over the part program        execution. These topics are later discussed in greater detail.        In an exemplary embodiment, all of the tasks that are not        associated with the user interface's run test tab can be        executed under the Uicontrol Thread.

According to one embodiment of the present invention, a nested statespattern is used. In one such embodiment, the state machine performs thestate activity of the parent state and then traverses the nested states.If the need for a transition out of the parent state is detected duringits state activity, no further activities of the nested states areperformed.

The data from the ballbar pipe can be permanently stored in a buffer(resamp_pipe_buffer) that contains the measurements over an entire testarc. The resamp_pipe_buffer may contain lower density data than theballbar pipe. This can be the case when ballbar pipe data is re-sampledfor the purpose of controlling the size of the dynamic test data file.

The lower density data can be created by performing a windowed-averageover sets of N_(resamp) raw data items, as illustrated in FIG. 21. Thedata container area of the resamp_pipe_buffer can be dynamicallyallocated to be capable of containing as many resampled data points asare expected to occur between the feedin and feedout events of thecontinuous motion test plus some amount of headroom.

Prior to each resampling operation, the number of items remaining in theballbar data-pipe can be compared to the resampling ratio N_(resamp). Ifthe number of items remaining is less than N_(resamp), the numberremaining can be stored as prev_num_rem, the average of the remainingitems can be computed and stored as prev_avg_rem, the ballbar pipe canbe emptied of its remaining contents, and an empty_ballbar_pipe( )method returns. The next time the empty_ballbar_pipe( ) method isinvoked, and the ballbar pipe contains data, the newest item in theresamp_pipe_buffer can be computed as described by the followingequation.${{next\_ sample}{\_ value}} = \frac{{{prev\_ avg}{\_ rem}^{*}{prev\_ num}{\_ rem}} + {\sum\limits_{i = 1}^{(N_{{resamp} - {{prev\_ num}{\_ rem}}})}\quad{{pipedata}\lbrack i\rbrack}}}{N_{resamp}}$

Once the initial resampled value is computed, the resampling processcontinues as usual. In the special case where the pipe contains fewerthan N_(resamp)-prev_num_rem data points, the values for prev_num_remand prev_num_avg can be updated, the data pipe emptied and the functionreturns. This can be accomplished by limiting the upper range of thesummation to the number of points in the pipe and the result of thecomputation is stored in prev_avg_rem, while the variable prev_num_remis updated by adding the number of points in the data pipe to theexisting value.

Upcoming discussions will describe how the data collection process in anexemplary embodiment begins some time in advance of the feedin move thattriggers collection of data to be retained. In the time prior to thistriggering event, the resamp_buffer can be populated via the resamplingprocess as described above. If the time between the beginning of thedata collection and the triggering event is exceedingly long, it cancause the resamp_buffer to fill up. If this occurs, then room for thenewest data can be created by discarding the necessary amount of theoldest data.

The resamp_buffer can be implemented as a ring-buffer linked-list. Asthe resamp_buffer is populated, a list-head-pointer can be updated topoint to the sample associated with the oldest measurement in thering-buffer and a list-tail-pointer points to the data item in thebuffer corresponding to the most recent measurements from the ballbarpipe. During the time in advance of the feedin trigger, the TIP FSM canscan the data from the most recent ballbar data pipe and check for theoccurrence of the triggering event. In one embodiment the “Empty PipeThread” checks for trigger conditions each time it runs in response to anew data event. Once the TIP FSM detects a valid triggering event, itcan retain a pointer to the buffer location associated with the event,two points are maintained—one for the most recent feedin trigger and onefor the most recent feedout trigger.

Exemplary embodiments of RTC and TIP finite state machines are furtherdescribed later herein. Some of the elements found in the FSM diagramswill now be described. As discussed herein, state machines are describedby diagrams that employ the symbols depicted in FIG. 22.

Each state is represented by a rounded corner box. The box can bepartitioned vertically into four sections. The top section contains thename of the state. The second section contains the state-entry actions(actions to be executed once when the state is entered via atransition).

The third section contains the state activity or in-state actions. Theseare the tasks associated with the current state that can be performedfor each invocation of the FSM. The state activity can be performedprior to any state-transitions. For the case of nested states, theactivity can be performed for both the parent and nested states, wherethe parent state's activity occurs first.

The bottom section of the state box contains the state-exit actions(actions to be executed once when the state is exited via a transition).Transitions from one state to another can be initiated by some stimuluscondition and are represented by a line with an arrowhead that points tothe state into which the system transitions. The transition line islabeled with an explanation of the stimulus followed (optionally) by aslash “/”, which is followed by the transition-action (an action toperform during the transition).

Facilities for distinguishing transition-actions from state-exit-actionsand state-entry-actions are included to avoid the need to duplicatestate-exit-actions on each transition out of a particular state, and toavoid the need to duplicate state-entry-actions for each transition intoa particular state. A diamond is used to indicate a conditionaltransition. A hollow circle represents the FSM's default-state (thestate of the FSM just after initialization or an FSM reset event). Asolid black circle with a number is used as a reference node. A solidgray circle with a number is a cross-reference to the reference nodewith the same number. Cross-references are used in the diagrams to avoidhaving too many arrows crossing.

In an exemplary embodiment, certain states are made to be relocatable tosatisfy employing several states, each with identicalstate-entry-actions, in-state-actions, state-exit-actions, andtransition stimuli. Rather than create a separate unique state for eachcase in which it is required, a relocatable state can be employed wherethe states to transition into in response to a stimulus are set up whenthe relocatable state is initially entered. A relocatable state isindicated in the diagram by a rounded-corner box with dashed linesrather than solid lines.

In an exemplary embodiment involving the ATB software, an ATB Run-Testpage (such as run-test page 30) includes a status-bar for the display ofa message that serves the dual purpose of: 1) providing prompts to theuser that describe what to do next to proceed; and 2) displaying thestatus while the FSM is in the process of completing some task. Many ofthe states in the FSM execute a function setstatus( ) at state entrytime. This function displays the particular message associated with thestate in the status bar. The particular message is not shown in the FSMdiagrams because of limited space, but the messages are provided in thetext descriptions of each state.

Machine motion might be involved while preparing for and executing aballbar test. Much of this machine motion can be accomplished bypopulating the control's part program MDI buffer with part programblocks. The function setmdi( ) can often be employed as a state-entryaction where a character string containing the specific part programassociated with the particular state is auto-generated (based on setupdata) and loaded into the MDI buffer. Explanations of the state specificMDI programs created by setmdi( ) are provided with the discussions ofthe individual states.

The setmdi( ) action might fail to populate the MDI buffer with a newprogram if the buffer is locked, which can occur, for example, when: 1)the program path control-state is such that the existing program cannotbe overwritten; or 2) the system is in the process of unlocking thebuffer in response to a change in control state. A number of states inthe RTC FSM involve an at-entry action of populating the MDI buffer witha new program. If the MDI buffer is locked at that point, then theattempt might fail.

In order to avoid this, transitions into a state with such an at-entryaction can be conditional upon a Boolean condition referred to as ateop(at end of program). In the context of discussions about the ATB FSMs,ateop is “true” when an attempt to clear the MDI buffer by invokingsetmdi( ) with a blank string is successful. The setmdi( ) action alsoresets the value of a waitinloop semaphore (process control data tableitem) for reasons discussed below.

State transitions are often the result of stimuli that indicate that theMDI program has thus-far executed beyond a certain block. The generationof these stimuli can be facilitated by including a block sequence numberfor each block in the auto-generated part program. The MDI programauto-generation method for a particular state provides the FSM with acollection of sequence numbers to represent specific locations in theprogram that have significance to the FSM. They may be used by the FSMto create a state transition stimulus when the sequence number of theblock currently undergoing execution (N_(SEQ)) equals or exceeds one ofthese values. N_(SEQ) is the notation used herein to represent thesystem name for the sequence number of the part program block currentlyundergoing execution.

Within the context of the RTC FSM, the state transitions typicallyfollow a pattern from a waiting-to-execute-mdi state to an executing-mdistate, to an instance of a WaitCycCompl state (described later herein),to a new waiting-to-execute state, etc. The condition N_(SEQ)≧N_(START)can be used to generate a transition out of a waiting-to-execute stateto an executing state. The transition out of the executing statetypically occurs when N_(SEQ)≧N_(STOP).

The transition out of the executing state typically enters aWaitCycCompl state, which waits for the control-state to be ateop andfor the MDI buffer to become unlocked so that it may be populated withprogram blocks upon entry of the next waiting-to-execute state. Withinthe context of the TIP FSM, a list of sequence numbers can be generatedduring part program action where the significance of each sequencenumber is understood by the FSM since the program is created by the FSM.The execution of the part program block that follows a given sequencenumber can be used to stimulate transitions within the TIP FSM as thecircular motion part program progresses.

Synchronization of the state of the FSM in response to the state of the,part program can be accomplished using sequence numbers N_(SEQ) asdescribed above. In certain instances, the realization of the ballbartest involves synchronizing the part program execution to the FSM state.One such example is now described.

In an exemplary embodiment, when the state of the FSM transitions due toa condition N_(SEQ)≧N_(START), the part program, at some point, suspendsand allows the FSM to complete its transition. Otherwise, the programexecution could proceed to completion, reset N_(SEQ) to zero and causethe FSM to be stuck waiting for a N_(SEQ)≧N_(STOP) stimulus conditionthat has already occurred and vanished. This situation can be avoided bycausing the part program execution to wait for the FSM to catch up byemploying, for example, a waitinloop mechanism.

The waitinloop can be accomplished by including blocks in theauto-generated MDI programs that can cause the part program to suspendat certain points. The suspended part program may be released by the FSM(running in a separate process) by incrementing the value of a variableshared by both processes (a semaphore). In the case of the A2100control, such variables are provided for interprocess communication andare referred to as process control variables. During program generation,infinite loops that are conditional on the value of a process controlvariable can be inserted at the locations within the part program wherere-synchronization of the part program to the FSM is desired.

Once the state transition of the FSM is completed, it can release the“suspend” on the part program, as it is now ready to detect the stimulusconditions. An example of a sequence of part-program blocks toaccomplish a suspend-loop is shown below. Code that auto generates theseblocks can be employed during program auto-generation.

The release condition in the part program can be that the value of theprocess control data semaphore exceeds its previous value, and therelease action of the FSM can be to increment the value of thesemaphore. The previous value of the semaphore can be maintained in thepart program context by copying the value of the process controlvariable to a local program variable in the block immediately followingthe suspend-loop, and making the condition to exit the suspend-loop thatthe process control data item have a value greater than that of thelocal variable. The value for the process control parameter can be setto zero at state machine initialization and reset if the FSM writes newblocks to the MDI buffer.

The first suspend-loop encountered in a program should functioncorrectly because the local variable is initialized to zero when it isreferenced for the first time. Referring to the code snippet below, theinterrogations of CURPOS_PGM(X) can be included to create await-for-steady state condition to help prevent the suspend-loop frombeing prematurely released during part-program lookahead.

Code Snippet 1. Example Of an NC Program Blocks Suspend-loop N100[#FOO]=[$CURPOS_PGM(X)] N100 (DO WHILE [$PROCESS_DATA(99)J]<=[#RELEASE])N100 G4F.1 N100 [#FOO]=[$CURPOS_PGM(X)] N100 (LOOP) N100[#RELEASE]=[$PROCESS_DATA(99)J]

The waitinloop(‘susp_cond’) method can be invoked during part programgeneration to generate the conditional statement for the particularprocess control data table entry selected. The release method ofwaitinloop, indicated by waitinloop(‘release’), can be employed by theFSM to increase the value of the appropriate entry of the processcontrol data. The release is usually applied from within a state whereit is known from the sequence number that the program is currently in orabout to enter a suspend-loop.

An additional method, waitinloop(‘init’), searches the process controldata table for an entry whose value is zero and stores the index of theselected table entry (e.g., as an FSM property). The waitinloop(‘init’)method can be invoked when, for example, a run-test page is entered. Thevalue of the selected process control data table entry can be returnedto zero by invoking waitinloop(‘restore’) when the run-test page isexited. Although the previous discussion used A2100 specifics for thecase of an example, the same concept with different implementationdetails can be employed on other CNC's with the same results.

The control state of the real time part program path is an item that canbe made available to the ATB that executes on a separate non-real timeprocess. The control state variables can be read prior to each recurrentinvocation of the FSM method for in-state actions. The control state canbe represented by a variable, e.g., S1NC_CONTROL, that serves toenumerate the state of the program execution process.

In the current context, the control state can be used in a test todetermine whether the part program is undergoing execution as arequirement for a transition stimulus. In the context of the RTC FSM,the control state can be represented as a Boolean termed incycle. In anexemplary embodiment, the incycle Boolean is “true” only if the firstpart program path is actively executing, is in a feedhold condition, oris stopped at end of block as in a single-block situation. Theterminology poll(ctlstate) is used in the FSM diagrams as shorthand toindicate an in-state action of testing whether or not the program is incycle as defined above.

Several states in the exemplary FSMs have transitions that can bestimulated by an event where the current state has persisted for aperiod of time greater than a maximum time period defined for the state.This is referred to in the FSM diagrams as a timeout condition.Transitions associated with a timeout condition typically involvegeneration of an alert. Timeout conditions can be detected by capturingthe system clock time as one of the state's at-entry actions andsubsequently, during the state activity, calculating the elapsed timesince state entry and comparing it to the state's timeout_timeattribute. States that have a timeout transition can also have anat-entry action of timeoutset where the time at state entry is capturedand the timeout_time is specified.

A purpose of a Run Test Coordinator (RTC) FSM according to one exemplaryembodiment is to automate the physical set up of, for example, a ballbartest, execute the test, collect data, save the data to a file, andperform an initial data analysis. For example, a complete test executioncan involve the completion of five steps.

Once again referring to an exemplary embodiment involving the ATBsoftware, each step can be represented by a button on the ATB Run-TestPage. Each button has an associated checkbox to indicate whether theparticular step has been completed. When a button is enabled, the usermay press it to perform the associated task. In certain circumstances, abutton is disabled where button presses are inhibited and the button is“grayed-out” to inform the user that the task associated with the buttonmay not be performed at that point in time.

The five buttons can appear vertically on the run-test page, such as inan order specified from top to bottom with self-descriptive labelsspecified. The buttons are identified in the FSM diagram by thespecified integer number.

-   -   1. “Move To Center”    -   2. “Secure Center”    -   3. “Move To Start”    -   4. “Start Test”    -   5. “Save Data”

The FSM diagram uses a shorthand terminology for certain state-entryactions that pertain to the enabled/disabled status of the buttons andthe checked/unchecked status of the checkboxes. The terminologyenable(n₁:n₂) is used to indicate that the buttons whose identifyinginteger are within the range of n₁ up to and including n₂ should beenabled. Similarly, disable(n₃:n₄) can be used to indicate an actionwhere buttons n₃ through n₄ should be disabled.

In one embodiment, buttons are enabled in such a way as to indicatethat, if the user chooses, a previous step may be re-performed, butsteps may not be skipped. As has been described in the previousdiscussion regarding run test page 30, in an exemplary embodiment,button presses are not required to move forward through the testsequence (e.g., they are used to repeat a step that has already beencompleted). A test can be indicated as having been successfullycompleted by causing the associated checkbox to be checked andvice-versa. This action is indicated in the FSM diagram by the notationsset_compl(n₁:n₂) and set_incmpl(n₃:n₄), which sets the checkboxesassociated with indices n, through no as checked and sets the checkboxeswhose indices are n₃ through n₄ as unchecked.

The exemplary RTC FSM enables and disables certain uicontrols found onthe run-test page as a function of the state. The following exemplarylist identifies which uicontrols have their enable/disable attributeassigned by the RTC state FSM, and describes how and why they might bemodified.

-   -   1. feedrate edit field—a feedrate edit field can be made        available on the run-test page to allow the user to run multiple        tests at multiple feedrates without having to return to a setup        page each time. The feedrate edit field can be uniformly enabled        as an at-entry task for RTC states except for the TestInProgress        state where it is disabled. These actions are not indicated in        the FSM diagram of FIG. 23 in order to save space. When the        state is AtStart, the test part program has already been        generated and is resident in the MDI buffer. User changes to the        feedrate when the state is AtStart are accommodated by employing        a process control variable in the program to represent a        variable feedrate. When the state transitions into        TestInProgress, the value from the feedrate edit field can be        copied to the appropriate entry of the process control data        table and further edits to the feedrate edit field can be        disabled.    -   2. perform-backlash-test checkbox—the RTC FSM can determine        whether or not a backlash test should be performed based on the        state of the perform-backlash-test checkbox. In an exemplary        embodiment, the ability for the user to change the checkbox        state is disabled upon entry of the AtStart state and the        TestInProgress state and is enabled upon entry of all other        states. These at-entry actions are not indicated in the FSM        diagram of FIG. 23 in order to conserve space. The disabling of        checkbox modification at entry of AtStart and TestInProgress        ensures that the state of the checkbox accurately reflects        whether or not a backlash test is being performed. The test-part        program can be loaded into MDI during entry into the AtStart        state. In an exemplary embodiment, the part program blocks to        perform the motion for the backlash test are generated only if        the backlash check box is checked. Thus, in one embodiment, the        user should not be allowed to change the state of the checkbox        at any time while the MDI buffer contains the test part program.    -   3. aggregation of ballbar calibration controls—in an exemplary        embodiment, ballbar length calibration may be performed or        cancelled from a limited number of RTC states. The ballbar        calibration process suspends execution of the RTC FSM, so it        cannot be allowed to take place while the part program is in        cycle or has an opportunity to go into cycle. These actions can        be strictly enforced by a mechanism to be described later. The        uicontrols associated with ballbar length calibration can be        disabled at entry of certain states to help prevent the user        from trying to perform a length calibration when it is known for        certain that the machine is in-cycle. The ballbar calibration        controls can be disabled upon entry of the states        MovingToCenter, MovingToStart, TestInProgress, TestAbort, and        WaitForCycComplete. In an exemplary embodiment, the        accessibility of the ballbar calibration controls is not        modified at entry of an AlertMessage state and is set to enabled        at entry of the remaining states in the RTC.

An exemplary RTC FSM diagram is depicted in FIG. 23. The purpose of areusable AlertMessage state can be to generate a window with an alertmessage for the user, and to wait for the user to acknowledge the alertmessage. In the RTC FSM diagram of FIG. 23, instances of the reusableAlertMessage state are represented by an “A” surrounded by a dashed box.

A diagram showing the actions and transitions for an A state is providedin FIG. 24. In FIG. 24, the state labeled “GenericRTCState” representsthe state from which a transition to a particular instance ofAlertMessage had originated. The text to display in the alert window canbe specified during the transition from GenericRTCState to AlertMessage.

Also during this transition, the AlertExitState can be specified. TheAlertExitState is the state to which the FSM transitions whenAlertMessage receives an acknowledge (ack) stimulus. The ack stimulusoccurs when, for example, a user presses an “OK” button in the alertwindow. In an exemplary embodiment, the alert window is destroyed as theAlertMessage state is exited.

The RTC FSM can also employ the reusable state W, which serves thepurpose of delaying a transition between two states until part programexecution has completed and the MDI buffer is write accessible. The Wsignifies WaitCycComplete. The W states in the exemplary RTC FSM diagramactually represents a two-state subsystem with a WaitCycComplete and anA state, as shown in FIG. 25.

In FIG. 25, the “Generic RTC State” indicates that WaitCycComplete isreusable and unique instances of WaitCycComplete have their own uniqueentry transitions. Transitions into WaitCycComplete involve specifying aCycComplExitState. The CycComplExitState dictates the state into whichthe system transitions when the W state is exited. The A state can becontained within the W subsystem to facilitate the generation of alertsthat, for example, suggest the user Feedhold and Data Reset when theprogram execution does not complete in the expected time frame.

The following at-entry actions can be associated with theWaitCycComplete state:

-   -   1) timeoutset(1): the timeout_time is set to 1 second as a        default. This can be sufficient because entries into        WaitCycComplete are typically stimulated by a program sequence        number that indicates that the program execution has reached its        final suspend-loop. For instances of WaitCycComplete where a        case-specific timeout_time is involved, the default may be        overridden by explicitly programming a timeoutset( ) during the        transition.    -   2) waitinloop(release): allow program execution to continue in        case it is currently in a suspend-loop.        The following in-state actions can be associated with        WaitCycComplete:    -   1) poll(ateop): check for the program not incycle and for write        access to the MDI buffer.    -   2) poll(time): check whether the time since state entry exceeds        the specified timeout_time.        There are no at-exit actions associated with the WaitCycComplete        state. Meanwhile, the following can be transitions out of the        WaitCycComplete state:    -   1) ateop: the MDI buffer is writeable and the waiting state may        be exited. The state into which is transitioned is specified by        the variable CycComplExitState that was defined when        WaitCycComplete was initially entered.    -   2) timeout: if the WaitCycComplete state has persisted for a        period of time considered unacceptable, the state transitions        into A (AlertMessage) with an alert asking the user to manually        complete program execution and unlock the MDI (such as with the        message “Hit Feed-Hold and Data-Reset to Continue”). The state        transitions back into WaitingForCycComplete when the alert is        acknowledged. If the user had not data-reset, then after the        timeout_time has elapsed the alert can be generated again. This        can continue until the user does what is asked.

The Idle state is the default state of the exemplary RTC FSM followinginvocation of the ATB software. It is also the state to which the FSM isset in response to a reset event. The reset event can occur when, forexample, the test-plane selection or setup data is changed. The Idlestate can provide a starting point from which the user can initiateexecution of the sequence of tasks involved in performing a test, suchas by pressing the button labeled “Move To Center”. The followingat-entry actions can be associated with the Idle state:

-   -   1) enable(1), disable(2:5), set_incmpl(1:5): reset buttons and        checkboxes to initial state    -   2) statusmsg( ): “Press ‘Move To Center’ to begin the test”

There are no explicit state activities or at-exit actions associatedwith the RTC Idle state. With respect to transitions associated with theRTC Idle state and, more particularly, a press(1)/check_setup( )transition, a user press of, for example, a “Move To Center” buttoncauses a transition into a conditional test. During the transition, theaction check-setup( ) performs test setup validity tests, such as thosepreviously described herein. If the setup is valid, the transitionproceeds to the state PreMoveToCenter. If the setup is found to beinvalid, the state transitions to an instance of the reusable stateAlertMessage (the reusable state labeled A). A alert message text can bedisplayed, such as one derived from the list of error codes returned bycheck_setup( ), and used to instruct the user to switch to, for example,a setup-test page to compose a valid setup (e.g., for the givencircle-plane).

PreMoveToCenter is a state in which the FSM waits for an indication(e.g., for the user to hit the cycle-start button on the machine'spendant or panel) to cause the machine axes to move to the circle centerposition specified in the setup data (e.g., by running a MDI program).The following at-entry actions can be associated with thePreMoveToCenter state:

-   -   1) enable(1), disable(2:5), set_incmpl(1:5): still working on        completing “Move To Center” task.    -   2) statusmsg( ): “Ensure Safe Path To Center, Loosen Center        Thumbscrew—Then Cycle Start”    -   3) setmdi( ): generates a sequence of machine coordinates        motions to move the axes to the setup data's circle center        coordinates. The feedrate employed can be the test setup data        item: traverse-feedrate. The center coordinates can be        approached by first programming a block to move to the        horizontal-axis and vertical-axis coordinates of the circle        center. This can be followed by a block to move to the        perpendicular axis coordinates of the circle center. Each block        in the MDI program contains a sequence number that is        incremented by one relative to the sequence number. of the        previous block. The motion blocks can be followed by a        suspend-loop. The suspend-loop can be included to prevent a        situation where the real time execution of the program races        ahead of the non-real time execution of the FSM and finishes the        program before the FSM's in-state action that queries the        sequence numbers has had a chance to execute and detect the        state-transition stimulus. The code snippet shown below presents        an example of a MDI program created by PreMoveToCenter's setmdi(        ) method for the case of an X-Y plane circle.

Code Snippet 2. Example of MDI Program to Move to Circle Center N1 G71G60 G90 G45 F4000 N2 G98.1 X300 Y150 N3 G98.1 Z200 N4[#FOO]=[$CURPOS_PGM(X)] N4 (DO WHILE [$PROCESS_DATA(99)J]<=[#RELEASE])N4 G4F.1 N4 [#FOO]=[$CURPOS_PGM(X)] N4 (LOOP)N4[#RELEASE]=[$PROCESS_DATA(99)J]

-   -   4) set(N_(START),N_(STOP)): N_(START) is set to the sequence        number of the first motion block of the MDI program (N_(START)=2        for the example above). N_(STOP) is set to be the sequence        number of the suspend loop (N_(STOP)=4 for the example above).    -   5) est_prg_time( ): computes a rough estimate of the time to        complete a move to center (e.g., at 100% feedrate override).        This time estimate can be used to determine an        order-of-magnitude value for the timeout condition for the        MovingToCenter state.

Poll(ctlstate) and poll(N_(SEQ)) state activities can be associated withthe PreMoveToCenter state. Accordingly, the control state and sequencenumber can both be determining factors for the generation of thestimulus condition for a transition out of PreMoveToCenter. Meanwhile,there are no at-exit actions associated with the PreMoveToCenter state.Furthermore, with respect to transitions from PreMoveToCenter, and moreparticularly N_(SEQ)≧N_(START) & incycle, the state can transition toMovingToCenter when the user hits cycle start and the program executionreaches the triggering sequence number.

The exemplary RTC FSM is in the state MovingToCenter when the MDIprogram (generated by the at-entry action of the PreMoveToCenter state)is incycle. The following at-entry actions can be associated with theMovingToCenter state:

-   -   1) enable(1), disable(2:5), set_incmpl(1:5): still working on        completion of the “Move To Center” task.    -   2) statusmsg( ): “Moving To Center”    -   3) timeoutset: The timeout_time can be set to be five times the        estimated time for the move plus, for example, thirty seconds.        Meanwhile, the following state activities can be associated with        the MovingToCenter state:    -   1) poll(N_(SEQ)): detect stimulus for transitions toward next        state    -   2) poll(ateop): detect stimulus for error condition where        program execution terminates prematurely    -   3) poll(time): test for timeout condition        With respect to an exemplary waitinloop(‘release’) at-exit        action associated with the MovingToCenter state, execution of        the MDI program is allowed to complete since an event to        stimulate transition out of MovingToCenter has occurred. The        following can be transitions from MovingToCenter state:    -   1) N_(SEQ)≧N_(STOP): when the sequence number exceeds N_(STOP),        the motion toward the center has completed and the state        transitions to the state “W” in the direction towards        SecureCenter. The W state acts as an intermediary to ensure that        the MDI buffer is unlocked (i.e., eop==“true”) before attempting        a setmdi( ) upon entry of the SecureCenter state.    -   2) ateop: This condition might occur if the user hits feedhold        followed by data reset. This condition stimulates a transition        to the reusable AlertMessage state in the direction of        PreMoveToCenter. The text for this particular transition can be        “Motion Toward Center Ended Prematurely: Press ‘OK’ To        Continue.” When the user hits the OK button in the alert window,        an acknowledge or ack stimulus can be created, and the state        transitions to PreMoveToCenter (where the user has a new        opportunity to complete the task of moving to the center).    -   3) timeout: if the timeout_time is exceeded, a timeout stimulus        can cause a transition toward an A state in the direction of        PreMoveToCenter. The alert message created during this        transition can contain the following text: “The move to circle        center did not complete in a reasonable period of time: hit        feed-hold and data reset to continue.’ An acknowledge of this        alert can cause a transition toward a W state before arriving at        PreMoveToCenter.

A purpose of a Secure Center state can be to provide the user with anopportunity to adjust the coordinates of the circle center relative tothe coordinates specified in setup data. For example, the user can beinstructed to adjust the axes positions by power feeding or handwheelingand/or to adjust the location of the ballbar center fixture on the tableuntil the magnetic cup picks up the center ball. If the user wishes forthe machine to return to the coordinates of the circle center as definedin setup data, the “Move To Center” button may be pressed. Once thefinal center is correctly established, the user can tighten the setscrewof the center fixture and hit cycle start to cause the FSM to capturethe current machine coordinates of the axes and transition to the nextstate.

Actions of an exemplary FSM in this case could include correctlyupdating the displays, detecting and responding to the cycle start, ordetecting and responding to a press of the “Move To Center” button,which might be the only button enabled. The cycle-start can be detectedin circumstances where the MDI buffer contains a program that consistsof just a suspend-loop. When the FSM detects that the program is incycleand has reached the sequence number of the suspend-loop, the state cantransition to a W state while simultaneously issuing awaitinloop(release). Additional description is found in the followingdiscussions of exemplary actions and transitions.

The following can be state activity associated with the SecureCenterstate:

-   -   1) enable(1:2), disable(3:5), set_compl(1), set_incmpl(2:5): the        move to center task is now complete. In an exemplary embodiment,        only the “Move To Center” button is enabled to allow the user to        return to the current active values for circle center        coordinates.    -   2) statusmsg( ): “Use Handwheel To Locate Ball In Magnetic Cup n        Tighten Screw, Then Cycle Start”    -   3) setmdi( ): The program loaded into the MDI buffer can simply        be a suspend-loop, such as the one in Code Snippet 1 where each        block in the program has the same sequence number.    -   4) set N_(START). Since each block in the MDI program has the        same sequence number, N_(START) can be set to this value.        Meanwhile, the following in-state actions can be associated with        the SecureCenter state:    -   1) poll(N_(SEQ)): test for partial condition for transition        stimulus    -   2) poll(ctlstate): test for partial condition for transition        stimulus

At-exit actions associated with the SecureCenter state includewaitinloop(release) where the waitinloop semaphore can be set to releaseregardless of which exit transition is taken. Meanwhile, transitionsFrom SecureCenter can include the following:

-   -   1) N_(SEQ)≧N_(START) & incycle: When execution of the MDI        program has begun, the state transitions to a conditional        transition node during which the current axis positions (in        machine coordinates) can be captured and assigned to the active        circle center coordinates for use in the upcoming test. In an        exemplary embodiment, the permanently stored circle center        coordinates are not modified by this process. The conditional        transition node performs a check_setup( ) validity test, as        previously described with respect to the discussion regarding        the maximum axis range for a circle within machine limits, to        assure that, given the modified center coordinates, the machine        envelope is still able to accommodate the circle test. If the        test fails, then the state can transition to an AlertMessage        state on its way back towards SecureCenter. The alert specified        during such a transition can indicate that the user's        modification of the center resulted in a set of test conditions        that cannot be accommodated by the machine and specify an        acceptable range of circle center coordinates. The        AlertExitState in this case is an instance of the        WaitCycComplete state to ensure that the MDI buffer is writeable        upon re-entry of the SecureCenter state. If the validity test is        successful, then the state transitions into an instance of        WaitCycCompl where the WaitCycComplExitState is set to        AtFinalCenter.    -   2) press(1): If the user presses the “Move To Center” button,        then the state transitions back towards PreMoveToCenter, but        first enters a WaitCycCompl state. In FIG. 23, the transition        enters reference node 1. This is depicted for the sake of being        able to cross reference all transitions into PreMoveToCenter        resulting from a press of button 1, and the actual FSM        implementation does not make use of a node 1 state.

The AtFinalCenter state can create an opportunity for a user to, forexample, handwheel or power feed the axes to bring the magnetic cup offof the center ball to a safe position (e.g., prior to a move to thestart position). A clearance move is not included as part of theexemplary program to move to start position as the FSM in thisembodiment does not know the orientation vector of the magnetic cup.Depending on the setup, the magnetic cup orientation vector may, forexample, be parallel to the tool vector, may be perpendicular to thecircle plane, or may be at some arbitrary angle. The exemplaryAtFinalCenter state transitions to PreMoveToStart when the FSM detectsthat the user has fed the axes a safe distance from the circle center.

The at-entry actions associated with the AtFinalCenter state caninclude:

-   -   1) enable(1:3), disable(4:5), set_compl(1:2), set_incmpl(3:5):        At this point, the tasks of “Move To Center” and “Secure Center”        are complete, but all others are considered incomplete. The        corresponding buttons can be enabled to allow the user to choose        a new center.    -   2) statusmsg( ): “Use power feed or handwheel to move the        magnetic cup a safe distance at least five millimeters away from        the center ball”    -   3) setmdi( ): The MDI buffer can be loaded with an empty string        to assure no consequences in case the user hits cycle-start.

Meanwhile, the state activity associated with the AtFinalCenter stateinclude poll(dfcc). With respect to this action, the distance fromcircle center (dfcc) can be computed from the instantaneous axispositions and the active value for machine coordinates of the circlecenter. The instantaneous axis position commands and following errorscan be sampled from the real time process prior to each recurrentinvocation of the current state's activities method.

The derived values for machine position can also be used to update thedisplays on a run-test page prior to each recurrent FSM invocation. Thevalue for dfcc can be computed for each invocation of the in-stateaction method by evaluating the square root of the sum of the squares ofthe individual axis distances from the active circle center.$\begin{matrix}{{dfcc} = \sqrt{\left( {{X(t)} - X_{circ\_ ctr}} \right)^{2} + \left( {{Y(t)} - Y_{circ\_ ctr}} \right)^{2} + \left( {{Y(t)} - Y_{circ\_ ctr}} \right)^{2}}} & (98)\end{matrix}$

In an exemplary embodiment, there are no at-exit actions associated withthe AtFinalCenter state. The following can be transitions from theAtFinalCenter state:

-   -   1) dfcc>5 mm: When the axes have moved, for example, at least        about five millimeters away from the circle center, the state        transitions to PreMoveToStart. If necessary, the user may        continue to feed the axes further from the center ball even        after the state transition without affecting the FSM.    -   2) press(1): When the user presses the “Move To Center” button,        the state transitions to the instance of WaitCycComplete at        reference node 1.    -   3) press(2): When the user presses the “Secure Center” button,        the state transitions to the instance of WaitCycCompl at        reference node 2.

The PreMoveToStart state serves as a place to wait for the operator tohit cycle start to begin execution of the MDI blocks to move the axes tothe test start position. The pattern employed for the transitions out ofPreMoveToStart can be identical to the pattern employed byPreMoveToCenter. The test start position is the location prior to the(e.g., 1.5 mm) feedin along the ballbar orientation vector.

At-Entry Actions associated with the PreMoveToStart state can include:

-   -   1) enable(1:3), disable(4:5), set_compl(1:2), set_incmpl(3:5):        The “Move To Start” button is enabled but the check box is not        checked to indicate that the move to start task is currently in        process. The Move To Start button may be pressed from this        state, but it results in no action.    -   2) statusmsg( ): “Ensure a safe path to the start position, then        hit Cycle—Start”    -   3) set(N_(START)), set(N_(STOP)): N_(START) is assigned the        value of the sequence number for the motion span and N_(STOP) is        assigned the value of the sequence number of the suspend-loop        blocks.    -   4) setmdi( ): The MDI program contains, for example, a linear        interpolation block to move to the machine coordinates of the        start position at the traverse_feedrate from setup data. The        motion span can be followed by a suspend-loop, such as the one        in Code Snippet 1. The start position can be computed from the        setup data and the active value for circle center. The start        position can be computed by first determining the unit-vector        that describes the ballbar orientation (direction cosines),        multiplying the unit vector components by the quantity ballbar        length plus 1.5 millimeters and then adding the circle center        components. The involved computations can utilize equations (92)        through (95).    -   5) est_prg_time( ): compute and store a rough estimate of the        time to complete a move to start for a feedrate override of 100%        when starting from the current position of the axes.        With respect to state activities associated with the        PreMoveToStart state, there can be poll(ctlstate, N_(SEQ)),        involving a check for a transition stimulus condition.        Meanwhile, there are no at-exit actions associated with the        PreMoveToStart state. Transitions from the PreMoveToStart state        can include:    -   1) N_(SEQ)≧N_(START) & incycle: Transition to the state        MovingToStart.    -   2) press(1): Return to the PreMoveToCenter state via the W state        at node 1.    -   3) press(2): Return to the SecureCenter state via the W state at        node 2.

The state MovingToStart serves the purpose of waiting for the machinemotion to complete. The pattern employed for state transitions out ofMovingToStart can be identical to the pattern employed forMovingToCenter. At-entry actions associated with the MovingToStart statecan include:

-   -   1) statusmsg( ): ‘Axes moving to start position’    -   2) timeoutset( ): The timeout condition can be set to be five        times the estimated time plus, for example, 30 seconds.

State activity associated with the MovingToStart state can include:

-   -   1) poll(N_(SEQ)): test for transition stimulus condition    -   2) poll(time): compute time elapsed since state entry        At-exit actions associated with the MovingToStart state can        include waitinloop(release), uniformly allowing program        execution to complete regardless of the particular transition        taken to exit the state. Meanwhile, transitions from the        MovingToStart state can include:    -   1) N_(SEQ)≧N_(STOP): This condition indicates that the part        program execution has reached the suspend-loop and a normal exit        transition toward the state AtStart is initiated. The transition        goes toward AtStart but first enters an instance of the W state        to ensure that the MDI buffer is unlocked to enable AtStart's        setmdi( ) at-entry method to execute properly.    -   2) lincycle & ateop: If this condition occurs before the normal        exit condition (N_(SEQ)≧N_(STOP)), then an event such as a        manual data reset or a servo fail might have occurred. In        response to this stimulus, the state transitions to an instance        of AlertMessage where the following text can be displayed: “Move        To Start Position Terminated Unexpectedly.” When the alert        message is acknowledged, the state transitions to PreMoveToStart        to give the process a second try.    -   3) timeout: If the MovingToStart state has persisted for a        period of time much longer than expected, the state transitions        to an instance of AlertMessage where the following text can be        displayed: “The move to start position is taking much longer        than expected: hit feedhold data reset.” Acknowledgement of this        alert by the user causes the state to transition to an instance        of W. If the user had not data reset, the W state can display an        alert message asking the user to do so. Once the ateop condition        is met, the state transitions to PreMoveToStart to allow the        user to try again. A timeout condition might occur if the        feedrate override is set to zero or if the user presses        feedhold.    -   4) press(1): Return to the PreMoveToCenter state via the W state        at node 1.    -   5) press(2): Return to the SecureCenter state via the W state at        node 2.

AtStart serves to provide a state where an exemplary RTC FSM waits forthe user to hit cycle start to initiate the test execution. At entry ofthis state, the user can be instructed to place the ballbar in themachine. The state also checks the current axes positions to ensure thatthe machine is at the correct start position. If the axes positionsdeviate from the start position, then the state transitions toPreMoveToStart. This mechanism simplifies the state machine by allowingtransitions from states where the axes are expected to be at the startposition to be into AtStart. At entry of AtStart, the MDI part programthat causes the motion for the test measurements can be created. Adetailed description of a MDI part program created at entry of AtStartwill also be provided later herein.

The following at-entry actions can be associated with the AtStart state:

-   -   1) enable(1:4), disable(5), set_compl(1:3), set_incmpl(4:5): At        entry of AtStart, the tasks of “Move To Center”, “Secure        Center”, and “Move To Start” will have been completed. Any of        these steps may be repeated by pressing the associated button.        The Perform Test task has not been completed, but its button can        be enabled to indicate that it is the next task.    -   2) statusmsg( ): ‘At Start Position—Place ballbar in machine,        then hit cycle-start’    -   3) setmdi( ): The MDI program to generate the motion for the        continuous motion test and the backlash test can be generated        from the setup data. Explanations of such programs and methods        used to generate them are further provided herein. The first        part of the MDI program can be a suspend-loop to allow the state        machine to turn on and initialize the ballbar.    -   4) set(N_(START)): The first sequence number of the MDI part        program is assigned to N_(START).        The following state activity can be associated with the AtStart        state:    -   1) poll(N_(SEQ)): check for the event to stimulate a transition.    -   2) poll(dfsp): The dfsp is the distance from start position. It        can be computed using the instantaneous machine coordinates of        the axis positions obtained from the real time process. The        start position can be the position of the axes just prior to the        feedin move, as defined in the PreMoveToStart state. The dfsp        can be computed as follows.        $\begin{matrix}        {{dfsp} = \sqrt{\left( {{X(t)} - X_{start\_ pos}} \right)^{2} + \left( {{Y(t)} - Y_{start\_ pos}} \right)^{2} + \left( {{Y(t)} - Y_{start\_ pos}} \right)^{2}}} & (99)        \end{matrix}$

There are no at-exit actions associated with the AtStart state. Thefollowing can be transitions from the AtStart state:

-   -   1) N_(SEQ)≧N_(START)/capture_test_conds, check_fdrate( ): When        the user hits cycle start, the program can execute a suspend        loop until the FSM executes a transition into a destination        state with a waitinloop(release) at entry condition. Detection        of the suspend loop's sequence number triggers a transition        toward the state TestInProgress. Before entering TestInProgress,        certain actions and tests can be performed, as indicated by the        conditional transitions in the FSM diagram. During the initial        transition, all the test conditions and a comprehensive        description of the control's configuration at that instant can        be captured. The information captured can include the test-setup        parameters, axis rates parameters, axis gains, path rates        parameters, axis compensation data, feedforward parameters, etc.        In addition, the test feedrate from, for example, an edit box on        a run test page can be captured and further edits to the edit        box can be disabled. The appropriateness of the feedrate with        respect to the setup data's overshoot arc can be checked using        the methods as previously described. Also at this time, the        estimate of trigger to trigger time t_(est) can be computed as        previously described and its value stored for later use. If the        checks indicate that the overshoot arcs are not large enough,        then the state can transition into an instance of AlertMessage        to indicate the problem to the user and to supply an estimate        for the maximum allowable feedrate for the current setup data.        When the alert is acknowledged, the state transitions to an        instance of W, from which the state transitions to AtStart when        an ateop condition is satisfied. Once the state has reached        AtStart, the user may reduce the feedrate or return to, for        example, a test-setup page to lengthen the overshoot arcs. If        the feedrate checks out, then the FSM attempts to turn on and        initialize the ballbar, such as previously described with        respect to a software interface to the ballbar. If ballbar        initialization is unsuccessful, then the state transitions to an        instance of AlertMessage, then to the instance of W, then to        AtStart where the user can try again to perform the test. If        ballbar initialization is successful, then the state transitions        into TestInProgress where the nested state machine can be run.    -   2) dfsp>0.05 mm: If the ballbar is not at the correct position,        then the state transitions to PreMoveToStart without any alerts.        This allows transitions where the FSM believes the axes might be        at start to be directed to AtStart. This potentially avoids an        extra cycle start by the user. It also protects against        situations where the axes are handwheeled or power-fed away from        the start position.    -   3) press(1): If the user wishes to return to the center        position, the MoveToCenter button can be pressed to transition        the FSM to reference node 1.    -   4) press(2): If the user wishes to re-secure the circle center,        the SecureCenter button can be pressed to transition the FSM to        reference node 2.    -   5) press(3): If the user presses the MoveToStart button, the        state transitions to an instance of W prior to entry into        PreMoveToStart, as indicated by reference node 3 in the diagram.        This button can be made available to allow the user to move to        the start position in case the user believes the axes may have        been accidentally fed a very small amount (e.g., less than        0.05 mm) since arriving at the start position.

An exemplary TestInProgress state contains its own nested FSM. Thedetails of an exemplary TIP FSM are later provided herein. The currentdiscussion is limited to at-entry and at-exit actions of such aTestInProgress state. The current discussion also describes stateactivity for the TestInProgress state that occur uniformly regardless ofthe nested state. For each recurrent invocation of the RTC FSM where thestate is TestInProgress, the uniform state activity of theTestInProgress state can be performed prior to the execution of thestate activity of the nested state. In an exemplary embodiment, theat-entry and at-exit actions of the TestInProgress state are executedonly once at the appropriate time as for any other RTC state.

The at-entry actions associated with the TestInProgress state caninclude:

-   -   1) set_compl(1:3), set_incmpl(4:5): At this point in the        procedure, the tasks “Move To Center”, “Secure Center”, and        “Move To Start” will have been completed successfully. In an        exemplary embodiment, the enable status of the buttons is not        modified directly because uicontrols are uniformly disabled        during test execution.    -   2) statusmsg( ): The status message can vary depending on the        nested state. For example, the status message just prior to        entry of TestInProgress can be set during the transition and        indicates that ballbar initialization had been successful.    -   3) disable(enabled): uicontrols on the run-test page whose        accessibility property is “enabled” can be identified and stored        in a list. The accessibility property of each of these controls        can be set to disabled. Additionally the buttons to switch, for        example, the page can also be disabled. The list of previously        enabled controls can be retained so that the FSM knows which        controls to re-enable when the test completes.    -   4) create(cancelbutton): The cancelbutton is a new uicontrol        that can be created at entry of TestInProgress. In an exemplary        embodiment, this button is the only enabled control and serves        to provide the user with the ability to manually cancel the        measurement test. The callback method of the cancel button can        be to invoke the FSM with a “cancel test” stimulus.    -   5) create objects: This action can create the resamp_pipe_buffer        object and initializes its attributes. This action selects a        resampling rate, computes the estimated measurements from feedin        to feedout, and allocates memory for the resamp_pipe_buffer and        the various run objects (e.g., clockwise run, counter-clockwise        run, backlash run). The size of the resamp_pipe_buffer can be        determined from the estimate of the trigger-to trigger time        t_(est) (as previously discussed), the ballbar sample rate        f_(s), and the resampling period N_(resamp). The resampling        period is an integer number of samples and can be selected at        state entry by computing the maximum sample rate that results in        a data buffer whose size is less than or equal to a maximum size        of Buffsize_(MAX). The parameter Buffsize_(MAX) can be selected        based on the speed with which, for example, the ATB process is        able to perform the data analysis and display tasks of the        analyze-data page. A value for Buffsize_(MAX) for a Pentium II        system, for example, where the performance remains reasonably        smooth is Buffsize_(MAX)=5000. The total amount of memory        allocated for the resamp_pipe buffer is the max_num_samps. The        computation of max_num_samps takes into account the tolerance on        the accuracy of the estimated trigger to trigger time t_(TOL)        and the size of the ballbar data pipe N_(PIPE) (which can be        specified by a ballbar interface definition). $\begin{matrix}        {N_{RESAMP} = {{round}\left( {0.5 + \frac{f_{samp} \cdot t_{est}}{{Buffsize}_{MAX}}} \right)}} & (100) \\        {{{max\_ num}{\_ samps}} = {{round}\left( {\frac{f_{samp} \cdot \left( {t_{est} + {2t_{TOL}}} \right)}{N_{RESAMP}} + N_{PIPE}} \right)}} & (101)        \end{matrix}$        The data container areas for each test arc (clockwise and        counterclockwise) can be dynamically allocated at this time to        be large enough to contain the samples associated with the data        arcs.

State activity associated with the TestInProgress state can include:

-   -   1) check for ballbar errors: The ballbar status can be        interrogated for error conditions at each invocation of the        in-state-actions with the RTC state being TestInProgress. At        this point, the ballbar status is expected to be powered up and        collecting data; any other state for the ballbar interface        object is considered incorrect. Additionally, the minimum        ballbar deflection from the resamp_pipe_buffer can be tested to        ensure that the ballbar is not compressed at or below its        minimum range. In an exemplary embodiment, a similar test for        maximum range is not performed because the ballbar is expected        to be extended beyond its maximum range at various times during        the test. If the number of measurements in the most recent        ballbar data pipe is equal to the maximum pipe length, then a        pipe overflow can be assumed to have occurred over the interval        and a discontiguous data error stimulus generated. In one        embodiment, monitoring of ballbar status may not be necessary        because the ballbar thread provides an error event that        initiates a call to the FSM with a ballbar stimulus. Likewise,        the empty pipe thread checks the ballbar range and invokes the        FSM with appropriate stimulus when the ballbar deflection falls        below the minimum or exceeds the maximum valid range limit.    -   2) update displays: During the period where the state is        TestInProgress, a run-test page can contain a display of the        instantaneous ballbar measurement. This display can be updated        by copying the most recent value from the data pipe to the        display object.    -   3) run nested FSM: In an exemplary embodiment, following state        activities that are executed uniformly regardless of the nested        state, the state activities specific to the nested state can be        performed by running the exemplary TIP FSM. Once the nested        state's activities are completed, the nested FSM ceases        execution. If the nested state activities result in a transition        of the nested state, then the steps to complete this transition        can be executed before the nested FSM ceases execution. In the        case of a transition of the nested state, the following sequence        can occur: 1) performance of at-exit actions of the original        nested state, 2) performance of any transition actions and 3)        performance of the at-entry actions for the new nested state.        The states and transitions of the TIP FSM are further discussed        later herein. If execution of the TIP FSM indicates that the RTC        state should transition out of TestInProgress, then the        transition is executed in the context of the RTC FSM.        The following at-exit actions are associated with the        TestInProgress state:    -   1) turn off ballbar: When the TestInProgress state is exited for        any reason, the ballbar can be turned off and uninitialized.    -   2) re_enable(disabled): The uicontrols that were enabled prior        to state entry and were disabled during state entry can be        re-enabled at state exit.    -   3) delete(cancel): The “cancel” button that was created at state        entry can be destroyed upon state exit.    -   4) delete objects: Delete any objects created for the purpose of        temporarily storing measurement data.        Transitions from the TestInProgress state can include:    -   1) test succeeded: This transition can originate from within the        nested TIP FSM when the entire sequence of tests has completed        without any error conditions detected.    -   2) test failed: This transition can originate from within the        nested TIP FSM. When an error condition is encountered during        execution of the TIP FSM, the TIP FSM can generate a        nested-state-specific alert message and transition to an        AlertMessage state in the context of the RTC FSM.    -   3) press(cancel): The callback method for the cancel button can        be to execute the RTC FSM's transition method with the        press(cancel) stimulus. The press(cancel) stimulus can cause a        transition to an AlertMessage state that displays, for example,        the message: “Test Cancelled By User—Hit feedhold and Data-Reset        to continue.” When the user acknowledges the alert message, the        state transitions to an instance of W. If the user had not        data-reset, then W can create an alert indicating that the user        should do so. Once the ateop condition is achieved, the state        transitions to AtStart.    -   4) ballbar error: A ballbar error detected by the TestInProgress        state's uniform in-state actions can result in generation of an        alert message to describe the particular error condition. The        RTC state transitions to an instance of AlertMessage.        Acknowledgement of the alert causes a state transition to        AtStart via a W state.

TestComplete is a state entered after successful execution of a ballbarmeasurement test. This state persists until, for example, a user pressesa task button or generates a reset event by changing the circle plane orby modifying the setup data on the setup page. At-entry actionsassociated with the TestComplete state can include:

-   -   1) enable(1:5), set_compl(1:4), set_incmpl(5): All tasks except        saving data have been completed. All buttons can be enabled.    -   2) analyze_data( ): At entry into TestComplete, the clockwise        and counter-clockwise data-arcs that have been extracted from        the trigger-to-trigger measurement by the TIP FSM can be copied        to a temporary data file. The captured test conditions, and        control configuration can be copied to the data file's header.        The data file can be automatically loaded to, for example, an        analyze data page with the associated data processing taking        place.        There are no state activities or at-exit actions associated with        the TestComplete state. Transitions from the TestComplete state        can include:    -   1) press(1:4) transition to the associated reference node in the        FSM diagram.)    -   2) press(5): transition to the SavingData state

The SavingData state can be entered from the TestComplete state when theuser presses a Save Data button, for example. At entry of SavingData, amodal dialog window can be created. This window can allow a user tospecify a data file name and to provide specific information about themachine, the test particulars, etc. The dialog can preload values intothe fields for the date, time, control serial number, and file name. Inone embodiment, the user may modify any of these. The dialog box canalso contain fields where the user may enter machine information andmiscellaneous notes. When the user presses save, for example, a datafile can be created that contains the recent data-arc data, atest-conditions header, and additional header information containing theinformation pertaining to the dialog box fields. Once the file saveoperation has completed, the Save Data checkbox can be checked. TheSavingData state persists until the user presses one of the buttons orcauses a reset of the RTC FSM.

An exemplary continuous motion program consists of the sequence ofblocks shown below. Each block in the described program contains aunique sequence number greater than the sequence number of the previousblock. One exception to this can include that each block in asuspend-loop can have the same sequence number, but that sequence numberis greater than the sequence number of the block before the first blockof the suspend-loop.

-   -   1) suspend-loop to wait for completion of ballbar initialization        and power-up    -   2) set modal states and feedrate    -   3) feed-in move along the ballbar orientation vector    -   4) counter clockwise acceleration arc    -   5) counter-clockwise data-arc first half    -   6) counter-clockwise data-arc second half    -   7) counter clockwise deceleration arc    -   8) feed-out move along the ballbar orientation vector    -   9) suspend-loop to wait for preparation for the clockwise move    -   10) feed-in move    -   11) clockwise acceleration arc    -   12) clockwise data-arc first half    -   13) clockwise data-arc second half    -   14) clockwise deceleration arc    -   15) feed out move    -   16) a suspend-loop to ensure an opportunity to detect N_(STOP)        before program execution terminates (or continues on to a        backlash test).

Motion blocks can be programmed in machine coordinates to avoidsituations where the circle center is affected by offsets and/or machineposition-set. Referring to the syntax of the programming language usedon the A2100 as an example, the circle can be programmed in machinecoordinates by, for example, moving the axes to the circle start pointin G98.1 mode with linear interpolation, and then programming a circleby using incremental coordinates for the circle end point (G91) andincremental coordinates for the circle center (G2.02 or G3.02). The dataarc can be divided into two halves such that the two separate blocksaccommodate the possibility that the data arc is 360 degrees.

In an exemplary embodiment, auto program generation accounts forcircle-plane, circle-direction and the ratio of ballbar-radius to circleradius. For the XY and YZ plane, a clockwise move can be generated byprogramming a clockwise arc (e.g., G2.02) and a counter-clockwise movecan be generated by programming a counter-clockwise arc (e.g., G3.02).For the XZ plane, the coordinates are left-handed (the clockwise movecan be generated by programming a counter-clockwise arc (e.g., G3.02)and the counter-clockwise move can be generated by programming aclockwise arc (e.g., G2.02)).

Circular arc blocks can be generated by copying the horizontal andvertical (incremental) coordinates for end-point and circle center tothe command position of the associated logical axis for the particularcircle-plane. The program generation algorithm can be summarized belowfor the case of both the clockwise and counter-clockwise moves.

In an exemplary embodiment, steps 1 through 4 (see below) are executedonly once. Steps 5 through 10 (see below) can be executed in sequencetwice; once for the counter-clockwise move angles θ^(ccw) and once forthe clockwise move angles θ^(clw). In the following discussion, thepseudo-code uses the syntax <variable> to indicate a string representingthe value of the variable between the “<” and “>” symbols, and doublequotes are used to indicate that the string is copied exactly.

1. Compute the ballbar orientation relative to the circle plane whileaccommodating the possibility of a circle-radius that is smaller thanthe ballbar length. This is represented by the in-plane component of theorientation-vector cos φ and the out-of-plane component of theorientation vector sin φ. $\begin{matrix}{{{\cos\quad\phi} = \frac{R_{Circle}}{L_{Ballbar}}};{{\sin\quad\phi} = \sqrt{1 - \left( {\cos\quad\phi} \right)^{2}}}} & (102)\end{matrix}$

2. Compute the circle angles for each of the circular motion spans inthe program for the counter-clockwise move. $\begin{matrix}{{{start}\quad{of}\quad{acceleration}\quad{span}\text{:}\quad\theta_{1}^{\quad{ccw}}} = {\theta_{DataStart} - \theta_{Overshoot}}} & (103) \\{{{start}\quad{of}\quad{first}\quad{half}\quad{data}\quad{arc}\text{:}\quad\theta_{2}^{\quad{ccw}}} = \theta_{DataStart}} & (104) \\{{{start}\quad{of}\quad{second}\text{-}\quad{half}\quad{data}\quad{arc}\text{:}\quad\theta_{3}^{\quad{ccw}}} = {\theta_{DataStart} + {\frac{1}{2}\theta_{DataRange}}}} & (105) \\{{{start}\quad{of}\quad{deceleration}\quad{arc}\text{:}\quad\theta_{4}^{\quad{ccw}}} = {\theta_{DataStart} + \theta_{DataRange}}} & (106) \\{{{end}\quad{of}\quad{deceleration}\quad{arc}\text{:}\quad\theta_{5}^{\quad{ccw}}} = {\theta_{DataStart} + \theta_{DataRange} + \theta_{Overshoot}}} & (107)\end{matrix}$

3. Compute the circle angles for the blocks in the clockwise move.$\begin{matrix}{{{start}\quad{of}\quad{acceleration}\quad{span}\text{:}\quad\theta_{1}^{\quad{clw}}} = \theta_{5}^{\quad{ccw}}} & (108) \\{{{start}\quad{of}\quad{first}\quad{half}\quad{data}\quad{arc}\text{:}\quad\theta_{2}^{\quad{clw}}} = \theta_{4}^{\quad{ccw}}} & (109) \\{{{start}\quad{of}\quad{second}\text{-}\quad{half}\quad{data}\quad{arc}\text{:}\quad\theta_{3}^{\quad{clw}}} = \theta_{3}^{\quad{ccw}}} & (110) \\{{{start}\quad{of}\quad{deceleration}\quad{arc}\text{:}\quad\theta_{4}^{\quad{clw}}} = \theta_{2}^{\quad{ccw}}} & (111) \\{{{end}\quad{of}\quad{deceleration}\quad{arc}\text{:}\quad\theta_{5}^{\quad{clw}}} = \theta_{1}^{\quad{ccw}}} & (112)\end{matrix}$

4. Generate strings for the modal interpolation type, circle plane,logical axes and circle center names that correspond to the horizontal,vertical, and perpendicular axes of the circle plane. $\begin{matrix}\left\{ \begin{matrix}{{{{XY}\text{:}\quad G_{plane}} = {``{G17}"}},{H_{ax} = {``X"}},{H_{ctr} = {``I"}},{V_{ax} = {``Y"}},{V_{ctr} = {``J"}},{P_{ax} = {``Z"}},{G_{intp} = \left\{ \begin{matrix}{{ccw}\text{:}\quad{``3.02"}} \\{{clw}\text{:}\quad{``2.02"}}\end{matrix} \right.}} \\{{{{XZ}\text{:}\quad G_{plane}} = {``{G18}"}},{H_{ax} = {``X"}},{H_{ctr} = {``I"}},{V_{ax} = {``Z"}},{V_{ctr} = {``K"}},{P_{ax} = {``Y"}},{G_{intp} = \left\{ \begin{matrix}{{ccw}\text{:}\quad{``2.02"}} \\{{clw}\text{:}\quad{``3.02"}}\end{matrix} \right.}} \\{{{{YZ}\text{:}\quad G_{plane}} = {``{G19}"}},{H_{ax} = {``Y"}},{H_{ctr} = {``J"}},{V_{ax} = {``Z"}},{V_{ctr} = {``K"}},{P_{ax} = {``Z"}},{G_{intp} = \left\{ \begin{matrix}{{ccw}\text{:}\quad{``3.02"}} \\{{clw}\text{:}\quad{``2.02"}}\end{matrix} \right.}}\end{matrix}\quad \right. & (113)\end{matrix}$

5. Generate the feedin block to move the axes linearly along theorientation of the ballbar toward the circle-center.

-   -   incremental axis motion in hvp system        h=−1.5·cos φ·cos(θ₁); v=−1.5·cos φ·sin(θ₁); p=−1.5·sin φ  (114)    -   block string        str=“G91G1”<H _(ax) ><h><V _(ax) ><v><P _(ax) ><P>

6. Generate the acceleration arc block.

-   -   incremental coordinates of arc end-point in h-v-p system        h _(end) =R _(Circle)·cos(θ₂−θ₁); v _(end) =R        _(Circle)·sin(θ₂−θ₁); p _(end)=0  (115)    -   incremental coordinates of circle-center in h-v-p system        h _(ctr) =−R _(Circle)·cos(θ₁); v _(ctr) =−R _(Circle)·sin(θ₁),        p _(ctr)=0  (116)    -   block string        str=“G91”<G _(plane) ><G _(intp) ><H _(ax) ><h _(end) ><H        _(ctr) ><h _(ctr) ><V _(ax) ><v _(end) ><V _(ctr) ><v _(ctr)>

7. Generate the first half data-arc block

-   -   incremental coordinates of arc end-point in h-v-p system        h _(end) =R _(Circle)·cos(θ₃−θ₂); v _(end) =R        _(Circle)·sin(θ₃−θ₂); p _(end)=0  (117)    -   incremental coordinates of circle-center in h-v-p system        h _(ctr) =−R _(Circle)·cos(θ₃); v _(ctr) =−R _(Circle)·sin(θ₂),        p _(ctr)=0  (118)    -   block string        str=“G91”<G _(plane) ><G _(intp) ><H _(ax) ><h _(end) ><H        _(ctr) ><h _(ctr) ><V _(ax) ><v _(end) ><V _(ctr) ><v _(ctr>)

8. Generate the second half data-arc block

-   -   incremental coordinates of arc end-point in h-v-p system        h _(end) =R _(Circle)·cos(θ₄−θ₃); v _(end) =R        _(Circle)·sin(θ₄−θ₃); p _(end)=0  (119)    -   incremental coordinates of circle-center in h-v-p system        h _(ctr) =−R _(Circle)·cos(θ₃); v _(ctr) =−R _(Circle)·sin(θ₃);        p _(ctr)=0  (120)    -   block string        str=“G91”<G _(plane) ><G _(intp) ><H _(ax) ><h _(end) ><H        _(ctr) ><h _(ctr) ><V _(ax) ><v _(end) ><V _(ctr) ><v _(ctr)>

9. Generate the deceleration-arc block

-   -   incremental coordinates of arc end-point in h-v-p system        h _(end) =R _(Circle)·cos(θ₅−θ₄); v _(end) =R        _(Circle)·sin(θ₅−θ₄); p_(end)=0  (121)    -   incremental coordinates of circle-center in h-v-p system        h _(ctr) =−R _(Circle)·cos(θ₄); V _(cir) =−R _(Circle)·sin(θ₄);        p _(ctr)=0  (122)    -   block string        str=“G91”<G _(plane) ><G _(intp) ><H _(ax) ><h _(end) ><H        _(ctr) ><h _(ctr) ><V _(ax) ><v _(end) ><V _(ctr) ><v _(ctr)>

10. Generate the feedout block to move the axes linearly along theorientation of the ballbar away from the circle-center.

-   -   incremental axis motion in hvp system.        h=1.5·cos φ·cos(κ₅); v=1.5·cos φ·sin(θ₅); p=1.5·sin φ  (123)    -   block string        str==“G91 G1”<H _(ax) ><h><V _(ax) ><v><P _(ax) ><p>

An exemplary backlash test part program causes the axes to move to anumber of locations at circle quadrant boundaries and, at theselocations, effects the axis motion associated with a backlashmeasurement. The measurements take place at each quadrant boundaryencountered along the data arc when traversed in a counter-clockwisedirection. The counter-clockwise motion begins and ends at the samelocations as the continuous-motion counter-clockwise test.

A suspend-loop can be inserted in the program after the final block thatis part of the ccw arc. This can be followed by a clockwise arc to movethe ballbar back to its start position (and to unwind a ballbar cord, ifnecessary). The final block that is part of the clockwise arc can befollowed by a suspend-loop.

The circle quadrant boundaries can be defined as the locations on thecircle whose counter-clockwise angle relative to the positive horizontalaxis is 0°, 900, 180°, and 270°. A test_point can be established foreach of these quadrant boundaries that is within the range or the dataarc identified in test setup data. For each test_point, a series ofmoves along the circle radial can be made to accommodate, for example,three ballbar measurements.

Prior to performing the first measurement, the axes move to a positionto ensure that the first measurement position is approached from thecorrect direction. The approach_loc coordinates can be along the circleradial that intersects the test_point a distance from the center that isgreater the R_(circle). The first measurement, meas1, for a particulartest_point can be made when the axes are at rest a distance R_(circle)from the circle center, where the location was approached with a singleaxis move originating at approach_loc.

The second measurement, meas2, for a particular point can be made at alocation along the quadrant boundary radial a distance somewhat lessthan R_(circle) from the circle center and can be approached directlyfrom the location of meas1. The third measurement, meas3, for aparticular point can be made at the same location as meas1, but isapproached from the opposite direction as meas1; the location for meas3can be approached directly from the location of meas2.

Dwell blocks can be inserted after each block that approaches ameasurement location to provide an opportunity for ballbar data to becollected while the axis is at rest. The blocks associated with themeasurements at a given test_point can be followed by a circular moveblock to the next quadrant boundary test point where blocks formeasurements at that point can be inserted. This exemplary motion isillustrated schematically in FIG. 26 for a case where the data-arc spansthree of the four quadrants. In the figure, the apparent spacing alongthe arc between the measurements at a single test_point is not intendedto represent any real relative displacement, but is included only tomake the figure readable.

Each block in the part program has a unique sequence number-greater thanthe sequence number of the preceding block. The sequence numbers can,for example, be used by the TIP FSM as transition stimuli to ensure thatthe proper ballbar data is collected and that it is attributed to theproper measurement. In order to accommodate this mechanism, the functionthat generates the backlash part program can populate and return a datastructure that contains program specific information for use by the TIPFSM. The structure identifies a hierarchy of information described bythe diagram shown in FIG. 27. The notation of FIG. 27 is UML (UnifiedModeling Language) and is used extensively in the field ofobject-oriented design.

The highest level in the hierarchy, bck_test, contains informationpertaining to the backlash test. Bck_test has an aggregation of the tworun_arc structures; one for a counter-clockwise directed arc and one fora clockwise directed arc. In an exemplary embodiment, measurements areperformed only during the counter-clockwise arc, and the bck_testcontains a clockwise run_arc to return the axes to their start position(e.g., while unwinding the ballbar cord).

The run_arc data pertains to information for a particular arc. Anexemplary run_arc has an aggregation of zero to four test_pointstructures. The test point contains information related to one of thequadrant-crossing test points. In an exemplary embodiment, the run_arccontains one or more test_point objects. The number of test_pointobjects owned by each run_arc can be equal and depend on the data arcstart angle and the data arc length.

According to one embodiment, if the run_arc contains only one testpoint, then a backlash estimate for only one of the horizontal orvertical axes is developed. If the run arc contains two or more testpoints, then a backlash estimate for both horizontal and vertical axescan be developed. The test points contain data to identify the axis towhich the backlash measurement applies. The test_points owned by thecounter-clockwise run_arc each contain an aggregation of threemeasurement objects. The test-points owned by the clockwise run_arcs donot own any measurements because they serve merely to form a paththrough which to move to unwind the ballbar cord.

In an exemplary embodiment, there is one measurement structure for eachof the locations meas1, meas2, meas3, at each test_point, as indicatedin FIG. 26. The data contained in each of these four objects can beutilized by, for example, a TIP FSM. Some of the data can be initializedduring the generation of the program blocks for the backlash test.Explanations of the individual data items are given below.

bck_test

-   -   1) backlash[2]: The estimated backlash for the horizontal=1 and        vertical=2 axis. The values can be computed by averaging the        backlash estimates for each test point that contains a        measurement of backlash for the particular axis.*    -   2) validest[2]: An indicator of whether the corresponding value        in the backlash array is valid.**

run_arc

-   -   1) direction: The direction of the run_arc +1=ccw, −1=clw in        terms of the machine coordinates (not necessarily the direction        in the circle plane coordinates: e.g. for YZ ccw        direction=−1)^(†)    -   2) program_string: A character string containing the motion        blocks for the run arc.^(†)    -   3) first_seqnum: The sequence number of the first block in the        run arc.^(†)    -   4) last_seqnum: The sequence number of the last block in the run        arc.^(†)

test_point

-   -   1) nominal_loc[3]: The logical axis machine coordinates (X Y Z)        of the test_point.^(†)    -   2) log_axis_ndx: The logical axis (X, Y, or Z) whose backlash is        measured at this test_point.^(†)    -   3) plane_axis_ndx: The circle plane axis (Horizontal or        Vertical) whose backlash is measured at this test_point.^(†)    -   4) backlash: The axis backlash measurement for this        test_point.**    -   5) valid: An indicator whether the value stored under this        test-point's backlash is valid.***    -   6) first-seqnum: The sequence number of the block that arcs to        the current test point.^(†)    -   7) last-seqnum: The sequence number of the block that arcs from        the current test point.^(†)

measurement

-   -   1) start_seqnum: The sequence number where ballbar measurements        begin.†    -   2) stop_seqnum: The sequence number where ballbar measurements        cease.^(†)    -   3) measnum: An indicator of which measurement the current        structure pertains to (e.g., 1, 2, or 3)    -   4) data[250]: A container for the sequence of ballbar        measurement samples.**    -   5) datasize: The number of entries in data[ ] that represent a        valid ballbar sample.**    -   6) mean: The average of the valid samples contained in data[        ].**    -   7) stdev: The standard deviation of the valid samples contained        in data[ ].**        notes: * set after completion of the backlash test    -   ** set during execution of the TIP FSM    -   *** updated during execution of the TIP FSM    -   † set during generation of the part program

An exemplary procedure for generating the program blocks andinitializing the values in the structures is described by the flow chartin FIG. 28. The flowchart shows how the program blocks to effect themotion depicted by FIG. 26 can be created. It is implied in theflowchart that each block has a sequence number that is equal to thevalue of N_(SEQ) at the time the block is created. The flowchart showshow the sequence numbers of blocks where measurements take place can bestored in the data structure. Motion blocks can be incremental and thecalculations for circular interpolation parameters and direction can bevery similar to those presented in the previous discussion regarding acontinuous motion program.

A measurement block is a dwell (a pause for a specified period of time)that is immediately preceded by a dwell and immediately followed by thedwell. In an exemplary embodiment, each of these dwells consumes theminimum amount of time possible. The dwell time for the measurementblock can be determined based on the desired number of contiguous datasamples to be taken and the expected latency between successiveinvocations of the FMS's state activities method. $\begin{matrix}{T_{dwell}^{measure} = {{\frac{N_{resamp}}{f_{sample}} \cdot N_{samples\_ desired}} + T_{Latency}}} & (124)\end{matrix}$

The dwell immediately preceding the data-collection dwell can serve toallow the axis enough time to unwind (i.e., reach a steady-statelocation). This amount of time may vary from machine to machine and canbe configurable. In an exemplary embodiment, the dwell immediatelyfollowing the data collection dwell provides a margin of safety in casethe interval latency is longer than expected.

An alternative solution involves one that does not require up-frontknowledge of the unwind time and the interval between invocation ofin-state-actions. The alternative embodiment might be more likely tominimize the total dwell time for each measurement. This embodiment isnot depicted in FIG. 26 or the TIP FSM diagram in FIG. 29 (see below),but could be implemented using a very similar underlying structure.

For example, dwell blocks can be replaced by a suspend-loop. When a TIPdetects that the program execution has entered the suspend-loop (e.g.,by polling sequence number), it transitions to a state that polls theballbar data. When statistical analysis of the ballbar data indicatesthat the axes have attained a steady-state condition, the TIP statetransitions to one where the ballbar data is collected. After collectingthe necessary samples, a FSM can issue a waitinloop(release) to allowthe program execution to proceed to the next measurement location. TheFSM state transitions to one waiting for the sequence number of the nextsuspend-loop. Such an embodiment is expected to be particularlyadvantageous for extended applications where many steady-statemeasurements at a variety of locations are involved and whereminimization of time for each measurement can be potentially important.

A TestInProgress finite state machine can run for each invocation of theFSM's state activity method when the state of a parentRunTestCoordinator finite state machine is TestInProgress. An exemplaryFSM diagram for a TestInProgress (TIP) is provided in FIG. 29. Theat-entry actions, state activity, and at-exit actions for aTestInProgress parent state have been previously discussed. Thediscussion will now focus on nested states, nested transitions, andactions that occur at-entry of a nested state, nested state activity,at-exit actions of a nested state, and transitions between nestedstates.

The exemplary TIP FSM can be divided into two sections: one for theperformance of a continuous motion test and one for performance of abacklash test. For example, a continuous motion test can be performedfirst and a backlash test can be performed second (if, for example, itsexecution was specified by the user). Individual nested states are nowdescribed.

The default state for an exemplary TIP FSM can be PreTest (e.g., thestate of the TIP FSM is reset to PreTest whenever the RTC statetransitions into TestInProgress). A purpose of PreTest can be toinitialize data for a continuous motion test and to check that theballbar is fully extended as expected. In an exemplary embodiment, whenPreTest is entered, the part program is suspended or will be suspendedat a location just before the feed-in move at the start of the (ccw orclw) continuous motion test. An exemplary PreTest state persists untilit is certain that the program is executing the suspend-loop.

The following at-entry actions can be associated with a PreTest state:

-   -   1) select dyn_run(Nrun), set N_(START) and N_(STOP): The        instance of the bbardata structure that pertains to the test        about to be run as indicated by Nrun is selected. The dyn_run        structure contains the value for the upcoming run's start        sequence number N_(START) that is the sequence number of the        suspend-loop that immediately precedes the feed-in block. The        dyn_run structure also contains the upcoming run's stop sequence        number N_(STOP) that is the sequence number of the suspend-loop        that follows the feedout block. The instance of dyn_run also        contains memory for storing the entire ballbar measurement and        fields that facilitate the data collection process.    -   2) statusmsg( ): “Waiting For Ballbar To Detect Feed-In Move”        In-state actions associated with the PreTest state can include:    -   1) poll(N_(SEQ)): The sequence number is the factor that        stimulates a transition out of PreTest.    -   2) min{D}: Maintain the minimum value for ballbar Deflection (D)        that has occurred since the PreTest state has been entered. This        can be done to ensure that the ballbar does not unexpectedly        enter its active range while the machine is at rest.

No at-exit actions are associated with the PreTest state. Transitionsfrom the PreTest state include N_(SEQ)≧N_(START). WhenN_(SEQ)≧N_(START), then the program has reached execution of thesuspend-loop and the state transitions to WaitingForTrigger. In anexemplary embodiment, the suspend-loop is not released until thetransition into WaitingForTrigger is complete.

A purpose of the WaitingForTrigger state is to wait for the feed-in moveto be detected by the ballbar. As the state is entered, the part programcan be suspended in a suspend-loop that precedes, for example, a shortdwell block followed by the block that effects the feed-in move. Atentry of the WaitingForTrigger state, the waitinloop semaphore can beincremented to release the part program from its suspended condition andthe part program execution proceeds to a short dwell block. The stateactivities test for the trigger event or a timeout event.

The following at-entry actions can be associated with theWaitingForTrigger state:

-   -   1) timeoutset( ): capture the time at state entry and select a        timeout time that is somewhat greater than the sum of 1) the        dwell time that follows the suspend-loop and 2) the estimated        time for the feed-in move to bring the ballbar within its active        range. According to one embodiment, the time estimate of the        second term will have been computed during program generation        during estimation of trigger-to-trigger time.    -   2) statusmsg( ): “Waiting For Ballbar To Detect Feed-In Move”    -   3) waitinloop(release): The part program is released from its        suspended state as the final at-entry task.        State Activity associated with the WaitingForTrigger state can        include:    -   1) check for timeout: Check whether the total time since the        WaitingForTrigger state has been entered exceeds the timeout        time.    -   2) check{D}: wait for ballbar deflection to enter valid        measurement range. As previously mentioned, the FSM need not        explicitly check for the triggering condition because the empty        pipe thread monitors for ballbar deflection transitions into and        out of range, and any deflection transitions result in        invocation of the FSM with the appropriate ballbar triggered        stimulus specified.        There are no at-exit actions associated with the        WaitingForTrigger state. The following can be transitions from        the WaitingForTrigger state:    -   1) timeout: If the state persists for a period of time longer        than expected, there is likely a problem with the test. The        timeout could occur when, for example, the ballbar never        triggers because it is incorrectly fixtured, the user hits feed        hold, the control is in single block mode, etc. The timeout        condition causes a transition to the TestProblem state with        alert text “The start trigger for ballbar data collection did        not occur within the expected period of time—the test is being        aborted—hit feed hold and data reset”    -   2) any{D}<D_(max)−ε: D_(max) is the value output by the ballbar        when it is at maximum deflection. When at least one of the        resampled ballbar deflection measurements from the most recent        data-pipe falls below D_(max) by at least some small amount (ε),        the ballbar has entered its active range and a trigger event is        considered to have taken place. The state transitions to a        conditional state where the integrity of the trigger can be        tested. During this transition, the data_start_ptr can be set to        point to the sample in the resamp_buffer that corresponds to the        first sample that met the triggering condition. The conditional        state can be exited via one of the two transitions listed below.    -   a) false trigger: The existence of a false trigger can be tested        by examining the section of the resampled buffer between the        data-start-ptr and the list_tail_ptr. If any of the measurements        in this section of the resamp-buffer have a value greater than        or equal to D_(max), then a false trigger can be assumed to have        occurred. The sensitivity of the condition for a false trigger        may be adjusted by varying ε (the trigger-hysteresis attribute        of the pipe_resamp_buffer object). A starting value for E can be        0.05 mm. If a false trigger is detected, the state transitions        to r TestProblem and the following alert can be generated: “A        false trigger has been detected—this can be due to a problem        with the ballbar fixturing or machine vibration. Hit feed hold        and data reset and try again. If the problem persists, try        increasing the value for the parameter: ballbar trigger        hysteresis.”    -   b) else/initialize num_meas If data in the pipe following the        initial measurement that stimulated the transition are in range,        then the state transitions to CollectingData. The number of        resamp_buffer samples between the data_start_ptr and the        list_tail_ptr (inclusive) establishes the initial value for        num_meas attribute of the resamp_pipe_buffer object.

A purpose of the CollectingData state involves keeping track of theelapsed time since the feedin move and to test for the occurrence of thefeedout move. At entry actions associated with the CollectingData statecan include statusmsg( ) (“Continuos Motion Test In Progress: Run <Nrun>of 2—To Abort: Press ‘Cancel’”). Meanwhile, state activity associatedwith the CollectingData state can include the following:

-   -   1) check {D}: Check the values for ballbar deflection that        correspond to measurement from the most recent data pipe for a        deflection that exceeds the maximum in-range deflection D_(max).        Alternatively, this task is performed by the empty pipe thread        and detection of a feed out event results in invocation of the        FSM with the TRIGOUT stimulus.    -   2) update num_meas: Add the number of samples added to the        resamp_buffer during the current interval to the existing value        for num_meas.    -   3) check (t_(meas)): Use the updated value for num_meas to        derive the total time since the start trigger: t_(meas) and        compare this value to the estimated value for trigger to trigger        time.        There are no at-exit actions associated with the CollectingData        state. The following can be transitions from the CollectingData        state:    -   1) any {D}≧D_(max): When any sample derived from the most recent        ballbar data pipe exceeds D_(max), the state transitions to a        conditional node to test whether the deflection is likely to        have been caused by the feedout move. The conditional node        computes t_(meas), the time since the feed-in trigger, by, for        example, multiplying num_meas by N_(resamp)/f_(s), and compares        this value to the estimated value for trigger to trigger time        t_(est). If t_(meas) is less than t_(est) by more than the        tolerance t_(TOL), then the state transitions to TestProblem,        otherwise the state transitions to WaitForTestCompl.    -   2) t_(meas)>t_(est)+t_(TOL): If this condition occurs before the        detection of the feedout move, then the state transitions to        TestProblem. The existence of this condition,        t_(meas)>t_(est)+t_(tol), indicates that a period of time        greater than the estimated test time has elapsed since the        feed-in trigger without the occurrence of the feedout trigger.        This could be caused by, for example, a problem with the ballbar        fixture, a feedhold press by the user, or a very large circle        center offset.    -   3) N_(SEQ)≧N_(STOP): If this condition occurs before the        detection of the feedout move, then the state transitions to        TestProblem. According to an exemplary embodiment, if the        sequence number is greater or equal to N_(STOP), the sequence        number of the blocks comprising the suspend-loop following the        feedout move, then the feedout move will have occurred, but the        ballbar will have not been extended. This could be caused by,        for example, a ballbar fixturing problem or a very large center        offset.

The WaitForTestCompl state can be entered upon completion of asuccessful continuous motion measurement. A purpose of this stateinvolves waiting for the feedout block to complete and for the controlto enter the suspend-loop that follows the blocks associated with acontinuous motion test. The transitions out of this state can be to:

-   -   1) PreTest: for the case where the first continuous motion test        (circle-plane-ccw) has just completed and the second test        (circle-plane-clw) is pending.    -   2) PreBacklash: for the case where the second continuous motion        test (circle-plane-clw) has just completed and a backlash test        is to be performed.    -   3) WaitForAtEOP: for the case where the second continuous motion        test has just completed and a backlash test is not to be        performed.    -   4) TestProblem: for the case where the state has persisted in        WaitForTestCompl for a period of time longer than expected.

Additional discussions regarding this state have to do with the actionstaken during a successful transition out of the state due to thecondition N_(SEQ)≧N_(STOP). One of the actions of this transition can beto extract data-arc data from trigger-to-trigger data contained in theresamp_pipe_buffer. The raw data can then be manipulated to obtain arepresentation of circle radius. Discussions of these topics areprovided later herein.

The PreBacklash state provides a state between the end of a continuousmotion test and the beginning of a backlash test. It can also provide amechanism to enable the system to perform backlash measurements alongboth the counter-clockwise and clockwise arcs. In an exemplaryembodiment, however, measurements are only taken along thecounter-clockwise-arc. The state transitions out of PreBacklash when thesequence number of the active block is greater than or equal to thesequence number of the first block of the pending backlash arc (e.g.,run_arc[Nrun].first_seqnum≧N_(SEQ)). In response to this condition, thestate transitions to BacklashIdle.

At entry of BacklashIdle, the current or upcoming test point andmeasurement objects can be identified, and the start and stop sequencenumbers to identify the program bracket where these objects areapplicable can be extracted. The state activity compares the most recentsequence number of the currently executing block to the start and stopsequence numbers for the measurement and test point, and initiates astate transition when appropriate. At-entry actions associated with theBacklashIdle state can include:

-   -   1) identify context: The run number N_(run) is known when the        state is BacklashIdle. The test_point index N_(testpoint) can be        determined by checking N_(first) _(—) _(seqnum) and N_(last)        _(—) _(seqnum) for the current test_point structure and        comparing this to the most recent value for the program active        block sequence number N_(SEQ). If N_(SEQ) is greater than or        equal to N_(last) _(—) _(seqnum), then the test point index        N_(testpoint) can be incremented by one, the current test point        structure can be re-assigned, the test point's N_(first) _(—)        _(seqnum) and N_(last) _(—) _(seqnum) can be extracted from the        new test point structure, and the measurement index N_(meas) can        be set to one. If the test point had not changed, then the        measurement index N_(meas) can be incremented by one. The        measurement start and stop sequence numbers N_(STOP) and        N_(START) can be extracted from the measurement structure        identified by N_(testpoint) and N_(meas).    -   2) compute_backlash(testpt): If the test point index had been        incremented during the identify context at-entry action, then        the backlash parameter for the previous test point can be        computed and stored in that test-point's data structure. The        backlash can simply be measurement[3].mean−measurement[1].mean.        State activity associated with the BacklashIdle state can        include poll N_(SEQ). The sequence number of the active part        program block can be tested to determine whether it is within        the expected range for the current test point and measurement.        It can also be tested to determine whether the active block is        the block in which the measurement should take place. There are        no at-exit actions associated with BacklashIdle

Transitions from BacklashIdle can include:

-   -   1) invalid N_(SEQ): The sequence number of the actively        executing part program block can be considered invalid if it is        not within the range from the current test point's first        sequence number to the current measurement's stop sequence        number (inclusive). That is, the current sequence number is        invalid if the following inequality is false:        test_point[N_(testpoint)].first_seqnum≧N_(SEQ)≧test_point[N_(testpoint)].meas[Nmeas].stop_seqnum.        This situation is most likely to occur for the case where        N_(SEQ) is too large, which could occur if the FSM is “starved”        for resources and falls behind the part program execution. The        invalid N_(SEQ) stimulus would likely be superseded by the data        discontiguous error initiated from the parent-state's activity        of testing for ballbar errors. The case of N_(SEQ) being too        small should never occur, but can be included for the sake of        robustness (e.g., to help identify a design or implementation        error). In any case, if the inequality evaluates to false, the        state transitions to TestProblem where the following alert        message can be displayed: “There has been a problem with the        synchronization between the part program and the backlash        measurements: please try the test again. If the problem persists        contact your service representative.” This stimulus is not        expected to occur if the waitinloop approach described with        respect to the autogeneration of a backlash test program is        employed.    -   2) N_(SEQ)==N_(START): If the sequence number is within the        expected range identified above, then it can be tested to        determine whether the part program block associated with the        measurement is currently the active block. If it is the active        block (N_(SEQ)==N_(START)), then the state transitions to        BacklashMeasuring.    -   3) N_(SEQ)≧N_(SEQLAST): When the part program sequence number is        greater than or equal the current run's sequence number,        run[N_(run)].last_seqnum, then the state transitions to        WaitForAtEOP where the FSM waits for the test part program to        perform the clockwise move (to unwind the cord) and for write        access to be gained to the MDI buffer. The transition in the FSM        shows how this can be accomplished by incrementing Nrun and then        testing it for greater than 1. This can be included in the FSM        to enable additional measurements for multiple run's to        accommodate extended application of the FSM beyond the scope of        the design discussed by example herein.

A purpose of a BacklashMeasuring state can be to copy data from theballbar pipe to the data buffer area in the measurement structure whilethe part program is executing the dwell associated with the measurement.There are no at-entry actions associated with the BacklashMeasuringstate. The following can be state activities associated with theBacklashMeasuring state:

-   -   1) buffer data: Data from the FSM's copy of the ballbar pipe is        appended to the end of the backlash measurement's data area.        When the measurement's data array is filled to maximum capacity,        this action ceases but the state remains BacklashMeasuring.    -   2) poll N_(SEQ): Check for the sequence number of the active        part program block to increase thereby indicating that the        measurement should end.

Meanwhile, the at-exit actions associated with the BacklashMeasuringstate can include check data. In an exemplary embodiment, the backlashmeasurement contains at least a specified minimum number of samplesrepresenting a steady-state constant axis position. The standarddeviation of the measurement data can be computed and compared to aminimum acceptable value. If the standard deviation exceeds the minimumacceptable value, then the measurement can be considered invalid. In anexemplary embodiment, the minimum acceptable value considers the effectsof axis vibrations and measurement noise (e.g., a good starting valuemight be 0.10 microns). If the machine axis has large vibration levels,then this value may need to be increased. A purpose of the test can beto discard measurements that take place when the axis has not yetreached steady-state. For axes with high levels of servo-inducedvibration, the value for the standard deviation threshold may involveadjustment.

Transitions from the BacklashMeasuring state can includeN_(SEQ)≧N_(STOP). When the sequence number of the active part programblock increases to beyond the stop sequence number for the currentmeasurement (e.g., the sequence number of the dwell following themeasurement dwell) then the state attempts to transition back toBacklashIdle in preparation for the next measurement. If the datacollected during the recent measurement is found to be invalid asdescribed above, then the state transitions to TestProblem

The WaitForAtEOP state waits for the part program execution to completeafter the final backlash measurement has been successful. According toan exemplary embodiment, when the ateop condition is encountered, theMDI part program will have completed execution and write access willhave been obtained to the MDI buffer. This condition stimulates atransition of the parent RTC FSM state from the TestInProgress toTestComplete.

When the ateop condition is encountered, prior to exiting the in-stateaction, the backlash for the horizontal and vertical axes can becomputed. The backlash value for the horizontal axis can be obtained byaveraging together the valid backlash measurements for the test pointsassociated with the horizontal axis and copying the result to the firstelement of the array in the bck_test structure's backlash field. Ifthere are no valid measurements, then the first array element of thevalid field in the bck_test structure can be set false; otherwise, it isset true. Similarly the second elements of the valid and backlash arraysin the bck structure can be populated with vertical axis testpoints.

The TestProblem state serves as a location where the TIP FSM waits forthe user to acknowledge an alert generated by some error condition fromwithin the TIP FSM. The text of the alert can be specified during thetransition into the TestProblem state. The callback method for thisinstance of the reusable AlertMessage can be a call of the RTC FSMmethod to transition to the instance of W indicated in the FSM diagram.

Data processing that can take place for each transition out of the TIPFSM's WaitForTestCompl state is described below. For example, adescription will be given on how a segment within a continuous motiontest's data buffer that corresponds to a data arc can be identified.

Each continuous motion test result (ccw and clw) contains the contiguousballbar data from the feed-in trigger to the feedout trigger. Since thefeedrate for the data arc is constant, the segment of the buffer thatcorresponds to the data arc has a length (in samples) that is equal tothe effective sampling frequency multiplied by the arc-length in mmdivided by the programmed feedrate. $\begin{matrix}{L_{data\_ arc} = {\frac{{R_{circle} \cdot \frac{\pi}{180}}\theta_{{data\_ arc}{\_ length}}}{\frac{1}{60}F} \cdot \frac{f_{s}}{N_{resamp}}}} & (125)\end{matrix}$

The start point for the segment of data to extract can be determined byconsidering the phase distortion effect of the position controller andthe difference between the estimated trigger to trigger time and theactual trigger to trigger time.

The conversion of raw data measurement to a circle radius adjusts themeasurement based on ballbar length and takes into account thepossibility that the ballbar length may be greater than the circleradius. When the ballbar is powered on, the calibration offset and theballbar scale factor can be specified such that the measurement dataplaced in the pipe by the ballbar driver contains adjustments for thesefactors. If the data indicates that the ballbar is not lengthcalibrated, then zero can be passed as the calibration offset. Theabsolute ballbar length measurement can be obtained by adding thenominal ballbar length to the data from the pipe.

If the ballbar is length calibrated, the resultant sum represents thetrue ballbar length for the measurement. Otherwise, it represents anapproximate length. This value (true or approximate length) can then beadjusted to account for the possibility of a non-zero cone angle by, forexample, multiplying the raw measurement of ballbar Displacement D, bythe cosine of the cone angle—the ratio of programmed circle radius tonominal ballbar length. $\begin{matrix}{{r(t)} = {\frac{R_{circle}}{L_{ballbar}} \cdot {D(t)}}} & (126)\end{matrix}$

Yet another capability of, for example, a run-test page such as run-testpage 30 could be ballbar length calibration. Ballbar length calibrationis implemented within the scope of the exemplary RTC FSM describedherein, but may take place in an exemplary embodiment regardless of theRTC state as long as there is no actively executing part program and aslong as the ballbar is turned off (i.e. the ballbar state isuninitialized).

An exemplary mechanism for accomplishing ballbar length calibration isshown in the FSM diagram in FIG. 30. The “H” surrounded by a circlerepresents the historic state of the RTC FSM (i.e., the state that wasactive when the FSM was interrupted to perform a calibration).Transitions into the historic state result in the performance of theat-entry actions for the specific state.

A calibrate button can, for example, be disabled at entry of variousstates within the RTC FSM and the press(calibrate) stimulus cannotoccur. In cases where the calibrate button is enabled, a user press ofthe calibrate button can cause the evaluation of a test to see whetherthe MDI part program is in an ateop state and whether the ballbar isoff. If the MDI part program is not in an ateop state or if the ballbaris not off, then the press(calibrate) action can be ignored (asindicated by returning to the historic state of the RTC FSM).

If the state of the MDI program is ateop and the ballbar is turned off,then the RTC state can transition to calibrating state. The historicstate of the RTC FSM can be retained during the transition. Upon entryof the calibrating state, the ballbar can be initialized, powered on,and configured with the correct scale, factor and a calibration offsetof 0. The ballbar can then be commanded to begin collecting data to itsdata pipe.

Also upon entry of calibrating state, the timeout time can be set tosome value that corresponds to somewhat more than the amount of time tocollect, for example, 1000 ballbar measurement samples at the fullsampling rate f_(s). The state activity of the calibrating state can beinvoked for each subsequent recurrent invocation of the RTC FSM from therun-test thread. The state transitions out of calibrating state and intoan alert state if a timeout condition or ballbar error occurs. Thetimeout condition occurs when the state has persisted as calibrating fora time period that exceeds the expected maximum.

The possibility of a ballbar error condition can be tested prior to eachinterrogation of the ballbar pipe. The alert state can transition to thehistoric RTC FSM state when the alert acknowledge is received. The statetransition out of calibrating that represents a successful calibrationmeasurement takes place after, for example, 1000 ballbar measurementshave been collected. During this transition, the ballbar offset can becomputed and a data integrity check performed.

If the data is found to be okay, then a length-calibrated status can beset to true and the state transitions to the historic state of the RTCFSM. If the data integrity test fails, then the state transitions to anAlertMessage state and the length-calibrated status can be set to false.Acknowledgement of the alert causes a transition into the historic stateof the RTC FSM.

A data validity test can involve computation of the standard deviationof the 1000 measurement samples. If the standard deviation is less than,for example, 0.01 microns, the data can be considered valid; otherwise,it can be considered invalid. The length calibration offset is the valuethat can be added to the raw data measurements to obtain an absoluteballbar length measurement. The calibration offset can be computed bysubtracting the average of the thousand points from the user-enteredvalue for the calibration fixture length. $\begin{matrix}{L_{calib\_ offset} = {L_{calib\_ fixture} - \left( {\frac{1}{1000}{\sum\limits_{i = 1}^{1000}\quad D_{i}}} \right) - L_{nominal}}} & (127)\end{matrix}$

Software, such as ATB software capable of running on, for example,XAcramatic Release 4 and/or Release 5 software, can be written using,for example, the Matlab programming language and employing capabilitiesof the Matlab math and graphics libraries. Standalone versions that donot require an installation of Matlab software can also be created, suchas by using a Matlab Compiler to convert the M-code to C-code.Workstation services layer-capabilities can be accessed through MatlabMex Files that act as an interface between the Matlab source code andthe C-code calling conventions of the services functions. Other softwareimplementations could use other languages and graphics libraries. In anexemplary embodiment, mathematical analysis, thread handling, machineconfiguration data management, and/or finite state machines can, forexample, be written in C or C++ and employ, for example, Win32 API.Meanwhile, and also in an exemplary embodiment, graphical userinterfaces can be written in Visual Basic and re-use, where possible,objects from existing set of controls (e.g., A2100 ActiveX controls).

One advantage associated with certain ones of the described exemplaryembodiments of the present invention involves providing a quick and easyto use method of determining machine compensation values. Thiscapability is expected to be useful to, among others, machine toolOEM's, control retrofitters, and machine tool end users. For example,and with respect to ballbar testing, a technician at an OEM runoff linemight utilize an embodiment of the present invention to use a series ofstandard test setups that are present at initial invocation (e.g., of anapplet) or are loaded in (e.g., from Files from the disk of a control).Accuracy statistics derived from the measurement data could be recordedto a file that may be permanently stored in the OEM's database. Thetechnician could indicate that the control should update itscompensation parameters to correct for any errors present.

Meanwhile, a higher level technician or engineer might utilize such anembodiment to design standard setups to use during runoff. The higherlevel technician could also design and perform non-standard tests toassess overall machine performance or to diagnose manufacturingproblems. In both cases, it can be desirable if a test is easy toperform with minimal expertise and take a minimal amount of time tocomplete.

In one embodiment, the total time to perform tasks for a major planecalibration using a ballbar should be less than three minutes. Thesetasks include fixturing setups, test execution, update of compensation,and second test execution to verify the compensation. Auto tuning ofcompensation may be also very useful to OEM's of lower cost machines whodo not perform a full laser calibration on their machines because of theadditional cost due to the length of time involved.

A control retrofitter might similarly benefit from the ease of use andquick test times. For example, a retrofitter could benefit fromauto-tuning aspects of certain embodiment of the present invention inthat a machine can be compensated with a minimal amount of effort. Forthe sake of comparison with an embodiment of the present inventioninvolving a ballbar test, it is noted that using a laser interferometerto develop data tables for bidirectional axis compensation and crossaxis compensation could take as long as several days of effort.

Still further, an end user might benefit from embodiments of the presentinvention involving ballbar testing in several ways. For example, theease with which a ballbar measurement can be obtained might encouragemaintenance teams to perform tests at regular intervals in order todevelop a record of the machine performance. This record could be used,for example, as part of a predictive maintenance program. Moreover, amachine tool operator could perform a ballbar test prior to themachining process. In this scenario, the machine could be compensatedfor its existing errors due to the thermal conditions at that point intime. Additionally, the range of test could be limited to the machiningenvelope of the job about to be performed such that the compensation isoptimized for a limited-range machining envelope. This could enable anend-user to produce parts with accuracy that surpasses the accuracyspecifications of the machine.

The foregoing descriptions of exemplary embodiments of the inventionhave been presented for purposes of illustration and description only.They are not intended to be exhaustive or to limit the invention to theprecise forms disclosed, and modifications and variations are possibleand contemplated in light of the above teachings. While a number ofexemplary and alternate embodiments, methods, systems, configurations,and potential applications have been described, it should be understoodthat many variations and alternatives could be utilized withoutdeparting from the scope of the invention. Moreover, although a varietyof potential configurations and components have been described, itshould be understood that a number of other configurations andcomponents could be utilized without departing from the scope of theinvention.

For example, several aspects of exemplary embodiments discussed hereinare based on a constraint that the data from a particular ballbar cannotbe directly synchronized to internal control data. If directsynchronization is possible then this data could, for example, beemployed directly to allow for a potentially more powerful and robustanalysis of data. For example, the axis feedback positions could be usedto compute an exact angle for each ballbar measurement (as opposed to,e.g., a technique of estimating the angle based on elapsed time and anemulation of the interpolator). The axis feedback measurement could alsobe used to obtain parameters for a more precise lost motion model thataccounts for the exact nature of a windup/backlash transient. Suchcapabilities might also simplify the exemplary RTC and TIP Finite StateMachines embodiments previously discussed by example herein.

Moreover, although certain ones of the exemplary embodiments previouslydiscussed herein were based around an existing Renishaw ballbar product,other embodiments of the present invention can be adapted with respectto, for example, other types of sensors, including other displacementsensors and other communication interfaces. For example, ballbars thatoutput data such as A quad B encoder signals could potentially provideexact synchronization between the ballbar and internal control data toprovide advantages such as those described above.

In addition, capabilities such as automatic derivation of proposals foradjustments of friction feedforward parameters can be included. Such acapability might address performance problems that arise due to thedifference between a friction transient and an ideal Coulomb frictionmodel assumed (e.g., by feedforward auto tuning). An iterative approachfor friction compensation adjustments has been outlined in thisapplication that identifies a basis for an algorithm that could beemployed to address such issues. Still further, the techniques andapproaches laid out in this application are expected to be largelyadaptable to other product lines of controllers, such as the Siemens 840and Sinumeric CNC controller product lines, for example.

One potential limitation of exemplary embodiments as described hereincan be that the compensation parameters derived from a ballbarmeasurement are only reliably applicable to regions of the machiningenvelope that are contained within the envelope of the ballbar testsused to derive the parameters. Typically, the results from threemeasurements (one for each of the XY, XZ, YZ plane) are used to developmachine compensation parameters. The range of motion associated withthese tests identifies a volume within the machine envelope where theparametric compensation is known to be applicable.

The entire machine envelope could be accurately represented by thecompensation parameters if the parameters could be derived based onmultiple measurements in each plane, where the combined area for eachmeasurement spans the entire machine envelope for the plane. Forexample, this could be done by simultaneously analyzing multiplemeasurements or by computing a weighted average of results fromindividual measurements. Such a capability might also involvemodification to the overall exemplary test environment previouslydescribed herein that is designed around the idea that a single(bidirectional) ballbar test is used to derive the compensationparameters for the test plane.

Axis angular errors (roll pitch yaw) may cause linear motion that may beinterpreted as squareness and straightness errors by the curve fittingused in certain exemplary embodiments disclosed herein. In analternative embodiment, for example, the affect of each angular errortype on the ballbar measurement could be quantified, and an expanded setof tests and analyses determined, to enable angular errors to bedistinguished from displacement errors and extract parameters for botherror types. For example, compensation software that compensates thedisplacement error effects of angular errors could be included.Compensation of this type would likely involve compensating an axisposition based on the position of the multiple linear axes (e.g., allthree).

The tasks performed by a user with respect to certain embodimentdisclosed herein may also be further automated by, for example, locatingthe physical ballbar test setup on a pallet that can be automaticallyloaded into the machine. A public interface (such as that of the A2100)could be used to generate any required “cycle start” presses. The toolholder containing the magnetic cup could be placed in the machine's toolmagazine.

A ballbar fixture on the pallet can contain the ballbar already attachedto the center ball at a pre-determined location on the pallet. The testsetup data can contain the parameters for this predetermined testlocation. The ballbar can be resting on a support post from which it canbe picked up by the tool. A hook can be used to drop the ballbar offback onto its support post once the testing is complete. One associatedadvantage of such an embodiment could involve facilitating the automaticscheduling of machine recalibration in a fully automated environment.

A ballbar also has potential for use as a sensor to enable automatedestimation of a machine's structural dynamics parameters. Such acapability might be required with, for example, auto-tuning parametersof a feedforward controller taking into account the deflection andoscillation of machine tool members whose positions are not directlymeasured by the position feedback sensor. For example, a ballbar couldbe used to estimate the stiffness and damping parameters of a machine'sstructural modes by employing it to measure the relative motion betweenthe machine tool's spindle nose and the workholding table. Thesemeasurements can be synchronized to measurements of the axis positionsand torques.

Modal parameters can be obtained by correlating the ballbar measurementsto the axis measurements taken under conditions of applied excitationsignals that contain energy of a wide frequency range. The excitationsignals can be generated by, for example, developing a part program thatoutputs a position command “chirp” that specifies interpolation of asmall diameter circle with successively increasing feedrate. Thediameter of the circle can be set to be somewhat smaller than the rangeof the ballbar. This situation can be summarized by FIG. 31. Matrixequations for estimating the axis parameters for an axis with a singledominant mode of vibration are also shown in FIG. 31.

Accordingly, in an exemplary embodiment, ballbar measurements aresynchronized to axis position and torque measurements. Synchronizationcan be accomplished by, for example, using a ballbar device whose outputcan be synchronized directly. For cases such as a serial ballbarinterface, synchronization might be accomplished during data analysis bymatching axis feedback and ballbar measurement data for the chirp timeslice that contains only low frequency energy.

The overall method employed by the exemplary embodiments disclosedherein to generate motion, collect data, and analyze data is alsoapplicable to other auto-tuning applications. For example, referring nowto FIG. 32, one embodiment of the present invention involves providing agraphical user interface (GUI) that enables a user to specify testparameters, manually change configuration data, view and analyzemeasurement/analysis results, and initiate a test sequence. Oneexemplary approach to accomplishing this involves an interface that cancontain test-specific objects such as those listed below:

-   -   1) GUI for providing the user access to the capabilities        specific to the test type    -   2) Program Generation Object to construct part program commands        to effect a desired machine motion    -   3) Setup Data Container for storing test parameters and checking        their validity.    -   4) Sensor Interface to manage any auxiliary sensors—such as a        ballbar device    -   5) Data Integration Object to merge and synchronize data from        multiple sources such as NC kernel data acquisition data, data        from axis drives, and data from auxiliary sensors.    -   6) Parameter Estimation Curve Fitting Object to provide        case-specific requirements for data analysis and parameter        extraction.    -   7) Test Execution Coordinator FSM to manage and respond to        inputs originating from a user, a RealTime kernel, and sensor        objects.

These objects can be arranged in one embodiment with the relationshipsroughly defined by FIG. 32.

Thus, it should be understood that the embodiments and examples havebeen chosen and described in order to best illustrate the principals ofthe invention and its practical applications to thereby enable one ofordinary skill in the art to best utilize the invention in variousembodiments and with various modifications as are suited for particularuses contemplated. Accordingly, it is intended that the scope of theinvention be defined by the claims appended hereto.

1. A machine tool, comprising: a) an actuator having an attachedmechanical member; b) a motion command generator adapted to producemotion commands capable of controlling the actuator; c) a compensatoradapted to compensate the motion commands based upon at least onecompensation parameter; d) a controller in communication with theactuator, the motion command generator, and the compensator, and adaptedto: i) receive an indication of a compensation parameter to be tested;ii) based on the compensation parameter to be tested, cause a signalassociated with a desired motion of the mechanical member to becommanded; iii) acquire control data associated with the signal; iv)acquire measurement data associated with actual motion of the mechanicalmember in response to the signal; v) analyze the control and measurementdata; vi) based on the step of analyzing the control and measurementdata, implement a value of the compensation parameter; wherein thecontroller is further adapted to provide a setup interface that enablesa user of the machine tool to design a test capable of causing thesignal to be commanded.
 2. The machine tool of claim 1, wherein thecompensation parameter compensates for at least one of a scale error, asquareness error, a straightness error, backlash, windup and friction.3. The machine tool of claim 1, wherein the controller is furtheradapted to prompt a user of the machine tool to setup a device capableof outputting the measurement data associated with the actual motion. 4.The machine tool of claim 3, wherein the controller being adapted toprompt a user of the motion control system to setup a device furthercomprises the controller being adapted to provide a test interfacecapable of aiding a user of the machine tool in setting up the device.5. The machine tool of claim 1, wherein the controller is furtheradapted to: cause the signal to be commanded again; acquire additionalcontrol data associated with the signal; acquire additional measurementdata associated with new actual motion of the mechanical member; andanalyze the additional data.
 6. The machine tool of claim 1, wherein thecontroller being adapted to analyze the data comprises the controllerbeing further adapted to: determine error from the data; associate theerror with a parametric model; and extract the value for thecompensation parameter from the parametric model.
 7. The machine tool ofclaim 6, wherein the controller being adapted to associate the errorwith a parametric model comprises the controller to be further adaptedto fit the error to a curve.
 8. The machine tool of claim 6, wherein thecontroller is further adapted to determine an ability of the parametricmodel to account for the error.
 9. The machine tool of claim 1, whereinthe controller being adapted to receive an indication of a compensationparameter to be tested comprises the controller being further adapted toprovide a setup interface that allows a test condition parameter to beentered, wherein the desired motion corresponds to the test conditionparameter that is entered.
 10. The machine tool of claim 9, wherein thecontroller is further adapted to validate the test condition parameter,wherein it is determined if the desired motion is appropriate based uponthe test condition parameter.
 11. The machine tool of claim 10, whereinthe controller being adapted to validate the test condition parametercomprises the controller being further adapted to describe a problemassociated with the desired motion if the test condition parameter isnot validated.
 12. The machine tool of claim 1, wherein the controllerbeing adapted to cause a signal to be commanded comprises the controllerbeing further adapted to: automatically generate a set of instructionsdesigned to cause the desired motion when executed; and execute the setof instructions.
 13. The machine tool of claim 1, wherein the controllerbeing adapted to analyze the data further comprises the controller beingadapted to provide an analysis interface capable of displaying ananalysis of the data.
 14. The machine tool of claim 1, wherein thecontroller being adapted to implement the value comprises the controllerbeing further adapted to: suggest the value for the compensationparameter; receive an indication with respect to acceptance of the valuesuggested for the compensation parameter; and if an indication withrespect to acceptance of the value suggested for the compensationparameter is received, implement the value of the compensationparameter.
 15. The machine tool of claim 14, wherein the controllerbeing adapted to suggest a value for the compensation parameter furthercomprises the controller being adapted to provide a diagnosticsinterface capable of allowing the value of the compensation parameter tobe adjusted.
 16. The machine tool of claim 15, wherein the controllerbeing adapted to provide a diagnostics interface further comprises thecontroller being adapted to provide a diagnostics interface capable ofdisplaying an error source and the value of the compensation parameteras suggested, and capable of receiving an indication of whether thevalue as suggested should be accepted.
 17. The machine tool of claim 14,wherein the controller being adapted to provide a diagnostics interfacefurther comprises the controller being adapted to display predictedaffects if the value is adjusted.
 18. The machine tool of claim 1,wherein the controller being adapted to implement the value comprisesthe controller being further adapted to: provide an adjustment interfacecapable of receiving changes to the value of the compensation parameter;and if received, implement the changes to the value of the compensationparameter in the machine tool.